The character is Willfully Weak. Maybe they're a Blood Knight who has more fun with hand-to-hand, or a Smug Snake that can beat you with their powers tied behind their back; both are more likely to pull out the big guns for a Worthy Opponent. Alternatively, simply detests their powers for some reason ("I don't want to rely on my superpowers too much" is a common one), and thus sticks to good ol' non-superpower methods like martial arts or mundane technology. One more possibility is that their powers are too strong. Being able to disintegrate people with a look (and having no "stun" option) is usually not acceptable to a hero with a Thou Shalt Not Kill moral code.
This trope was also an inadvertent side-effect of Coconut Superpowers in old shows and serials due to lack of the special-effects technology to properly depict fantastic powers.
Compare Boxing Lessons for Superman, where trained fighting is an optional extra the character takes to help with extreme circumstances; and Guile Hero, where they prefer trickery to outright confrontation. This trope can overlap with Empowered Badass Normal, if the character still needs to mostly rely on previous fighting skills post-upgrade.
NOTE: A De-powered character such as a Henshin Hero only counts if they can voluntarily switch them on and off at will and keeps them shut off most of the time, in which case they're Willfully Weak. The character must have a choice whether or not to (try to) utilize their powers to qualify for this trope, or otherwise have powers which have no bearing on combat that can't be realistically replicated (such as Longevity, which would remove them from Badass Normal territory). Also, if Everyone Is a Super then fighting with powers is normal — in such a setting, someone who can't or has reason to avoid doing so is actually a Handicapped Badass.
Alucard from Hellsing is a vampire with supernatural powers. Despite that, he prefers to simply shoot his enemies with custom-made guns rather than use his inhuman abilities, mostly because he considers the majority of his opponents not worthy of using his superpowers against them. (Disregarding the fact that a real "normal" firing those guns would easily cause arms to be broken by recoil.)
The titular character of Rune Soldier Louie is a mage, yet he prefers to fight with his hands, using his famous "LOUIE PUNCH!". He is called out on it multiple times.
Similarly, Tsuchimikado Motoharu can only use magic in limited amounts as it risks killing him (which his esper ability, Auto-Healing, only partly mitigates). Therefore, he relies on hand-to-hand combat, and is even better at it than the protagonist.
Non-violent example, but Mikoto's friend Kazari Uiharu is a level 1 esper, who is also a genius computer hacker known in certain circles as "the Gatekeeper." For a long time, it was assumed that her power boosted her hacking abilities in some way. Eventually, we found out her power is temperature regulation—which, obviously, has absolutely nothing to do with computer hacking. She's just that good.
Mii Konori fights using martial arts because her power is X-Ray Vision. Her ability is used to scan opponents for concealed weapons.
Zenkichi from Medaka Box only has passive powers Nullifying coincidence and Parasite Seeing, so he mostly still has to rely on his savate abilities in fights.
Golden Lion King from Rising X Rydeen has not choice but to rely on his hand-to-hand combat skills as his superpower which is that he can glow when he roars is useless in most situations.
One Piece: While Luffy can be very creative with his Rubber Man powers, he's still capable of fighting with fisticuffs, especially when facing Lucci.
The Immortal Iron Fist for the most part relies on his martial skill in combat, using his Iron Fist ability mostly as a finishing move since it was too damaging to use against mooks, and initially left him so tired he couldn't overuse it.
Black Canary from the DCU may be the single best example. Her only superpower (sonic scream) is rarely used, and she's probably gone on 15-20 issue streaks of not using it at all, but she's a martial artist good enough to give Lady Shiva pause.
Combat training is standard for the X-Men, since many of them have powers that are either not specifically combat-oriented or too dangerous to utilize except as a last resort. The most famous examples are probably Cable and Bishop, two time-displaced soldiers with vast powers who normally just use guns.
Cable's father Cyclops, on occasion, due to his power being similar to Canary's in effectiveness; great for big robots, dangerous for close quarters. Sometimes they'll enforce this by him being temporarily powerless.
Storm has the same issue as Cyclops with her powers (weather control is great outdoors, but doesn't work too well when in confined spaces underground), and a past as a street thief before her powers manifested, so she's able to use her fists pretty well. As demonstrated when she challenged Cyclops for leadership of the X-Men - while depowered - and beat his ass into the ground.
Hitman's Tommy Monoghan pretty much stopped using his powers entirely after the first twenty issues or so, getting by on charm, badass, and luck.
Spawn has super powers, but prefers not to use them, since they're granted by hell and each use further damns his soul.
Aquamanused to be like this, before writers began upgrading the crap out of him to counter the whole "LOL, Aquaman" meme. Nowadays he can just summon every shark in the ocean or a mountain sized tidal wave to crush you dead.
Daredevil has an element of this. He is technically a superhuman since he has Super Senses but those alone aren't enough to beat the baddies so he has elite martial arts skills in addition to his powers.
Bart Allen has a throwaway line where he calls himself "Batman's protege", using said training to briefly fight back against the Rogues despite being depowered before finally being overwhelmed by their numbers.
The Democratic Republic of Acturus from The TSAB Acturus War mostly uses its magic as passive adjuncts to mundane weaponry and equipment, such as enhancing the power of nukes or teleportation to auto-load tank rounds. Only the Assault Troopers directly use offensive magic. This is in contrast to the TSAB, where even the Redshirt Army's standard rifles fire mana bolts.
Invoked in Peter's fight against the Rhino where his use of tools such as a gun rather than relying on his Hydra powers let him win.
Also invoked in chapter 55 where he has to refrain from using Hydra so as to not break his cover in front of some Thunderbolts.
Gandalf in The Lord of the Rings only used his magic a few times, often preferring to go after opponents with his sword and staff. This is more or less true of the original novels as well but it is emphasized more in the movies. According to Word of God, Peter Jackson was not a huge fan of seeing magic onscreen and preferred Gandalf to be more physical.
Neo from The Matrix trilogy mostly used his kung-fu skills but had the ability to manipulate the Matrix, allowing him the ability to fly, telekinesis, and other powers that he only pulled out when he needed them.
In The Wheel of Time, Rand Al'Thor is the most powerful channeler in the world, but for the first half of the series prefers to use a sword since the male half of the One Power is tainted and he's having serious How Do I Shot Web? and Magic Misfire issues.
In the Warrior Cats series, out of the three cats with superpowers, only Lionblaze's are useful in battle (he can't get hurt in a fight). Jayfeather's power is to enter other cats' dreams (though he doesn't fight anyway, since he's a medicine cat), and Dovewing's power - Super Senses - is actually a hindrance in battle because the amount of noise and scents confuses her.
Richard of the Sword of Truth series has phenomenal wizardly might, but relies almost exclusively on his sword for combat, primarily because he has no bloody clue how to get his magic to work at will. This is a borderline example, since that sword is itself magic.
Fate/Zero: Kiritsugu definitely has the capacity to be a magus, but instead on relies a lot more on good ol' human technology like firearms and explosives, treating his magecraft abilities as simply yet another tool to be used, which he usually only uses to support his mundane abilities and devices in a largely indirect manner, unless they prove to be more efficient than mundane methods (e.g. giving himself Innate Night Vision) or capable of doing something useful to his purposes that mundane alternatives cannot (e.g. Bounded Fields). This serves him well, since almost all Magi despise the idea of depending on muggle-made technology, making it a blind spot to them whose true potential they consistently underestimate. A prime example is at the start of Volume 2 (Episode 6 in the Animated Adaptation), which sees Kayneth being way too prideful, thinking that nothing would be able to invade his multi-layered magic fortress. In response, Kiritsugu has his colleague Maiya rig the building itself with explosives, thus rendering all of Kayneth's careful preparations useless.
In Young Jedi Knights, Tenel Ka doesn't use the Force if she can achieve her goal with athletic skill.
Live Action TV
The Power Rangers regularly take out the Mooks without bothering to morph, as part of the series tradition of not escalating a battle beyond what's necessary. Keep in mind, though, that several teams have special abilities that they can and do use even while unmorphed, which disqualifies them for this trope.
A flash game fan remake of Kung Fu has a special ability you unlock after beating each of the first three stages, all can lay waste to every mook on the screen. When you get to a boss fight, the main character refuses to use these specials, on the basis that he can kick their butts without em. Considering who he's fighting in stages 2 to 4, that's a hell of statement to make.
Many primary spellcasters in Dungeons & Dragons 3.5e can hold their own quite well in a fight due to respectable hit point totals, equipment proficiencies and base attack bonuses. Among these are clerics, druids, warlocks, favored souls and spirit shamans.
A few advanced classes from the d20 Modern settings:
Urban Arcana: The thrasher (Having an impressive damage reduction coming from magic, i.e. Super Toughness), the Speed Demon (A pilot with supernatural enhancements to his/her piloting skills), the Acolyte thanks to Vancian Magic (he can break out powerful powers, but they have a very limited number of uses per day, so he either saves them or is forced to fight like a normal when he runs out).
Agents of PSI: The Battle Mind, whom can mostly enhance his natural fighting prowess with psionics, along with a few more exotic powers.
In Mage: The Ascension, it's possible to load up your character with combat stats and skills despite being able to use Magick. If the character doesn't have the Spheres or can't figure out a way to avoid Paradox, it's often advisable use conventional weapons in a fight rather than Magick,
Similarly in Mage: The Awakening. The risk of Paradox increases with each vulgar spell one casts in a scene, meaning that someone who relies on on throwing around fire and lightning in a fight will either be forced to take damage from mitigating the Paradox or have their spells go hilariously wrong. It's often better to use self-enhancement magics and/or enchanted weaponry.
Superhero School Whateley Academy in the Whateley Universe actually encourages this. One of the teachers for the aikido classes is sensei Tolman, who is a Type 3: a black woman with superstrength and mental attack powers, who just prefers using aikido. And Aquerna is a Type 1: she has squirrel powers, so she is stronger and faster than a baseline human, which means she is considered a campus joke. She has been learning aikido, Le Parkour, and martial arts weapons.
Many superheroes and supervillains in Worm learn how to fight this way, but Armsmaster, Grace, and Grue in particular tend to use their powers to augment their essentially-human martial art skills, and Victor uses his power to gain human martial art skills.
Amon from Legend Of Korra fight primarily using chi blocking, which is a non-supernatural fighting style designed to counter bending. He also has the ability to remove someone's bending, courtesy of being an incredibly powerful Waterbender who can Bloodbend any time, no full moon needed, and can do so with a Death Glare alone, but that isn't the kind of thing that he can do while being attacked. His ability to avoid all bending attacks makes him one of the most intimidating characters in the franchise, but he's also using his psychic bloodbending to help him throw off his opponents' movements, giving him a nigh unbeatable edge. Technically, this means that he really isn't fighting like a normal.