When characters become unlikable, usually unintentionally, the people responsible for creating that character or judging its fate tend to go in different directions. They might do something to make the character likable. Or, they might align with the audience and join in on the disdain. But, sometimes, often to the detriment to the audience and even at their own peril, they embrace the character and basically tell the audience they have to like this character.
As stated on the main topic, the Creator's Pet has to fill a complete set of criteria in order to be viewed as such (i.e. the fans hate them, but the writers and/or producers don't, and as such, they'll be crammed into scenes). It's pretty clear what makes a Creator's Pet, but how does a Creator's Pet come into existance? How can they wind up as one of the worst possible designations? There are quite a few examples as to how and why, and it all boils down to the audience and the scenario, although these can be universal.
The Pet is a manifestation or a representation of the viewpoints of the writers themselves, which makes them comfortable. However, their views make them a moral or philosophical minority compared to the audience in general, which is guaranteed to result in complaints. A more extreme version of this is that the Pet is an Author Avatar, in which it represents just one individual.
The Pet is a Moral Guardian or a Soapbox Sadie, to which they rally against just about anything. The arguments in these cases isn't that they shouldn't be objective, but rather that they tend to be intolerant to other opinions and merely badger the other characters endlessly until they cave in. It's worse when the Pet is "right" about their opinion, and they often are.
The Pet has immunity to their surroundings. For example, in a Crapsack World, most people tend to be outright failures, but a Pet always seems to succeed. And, even if they feel that the Pet can't win, the worst that happens is that they finish second. In summary, bad things can happen to most characters, but the Pet doesn't take any brunt, and in the off chance they do, it's a terrible tragedy. Ironically, this typically goes hand-in-hand with the next example...
The Pet is also a Canon Sue. The audience sometimes tends to associate Canon Sue with Creator's Pet, and although that isn't always the case, if a show intentionally shows that a character is "flawed" just because, rather than deconstruct the issue, it's merely an attempt to get the viewer to feel sorry for their problems.
The Pet is directly related to the people in charge in real life. This is exclusive to Professional Wrestling, which is arguably the most obvious (and notorious) depiction of the Creator's Pet in action. After all, promotions over the decades have often put the lead promoter or booker and/or their relatives as champions and part of main event storylines, often to the detrement of other performers and the fans themselves. In plenty of these well-documented cases, the Pets are given more than what they deserve or can contribute.
Once in a great while, Depending on the Writer results in a decent character winding up as a Creator's Pet. Such was the case for Allenby Beardsley in the manga readaption of Mobile Fighter G Gundam, who is a likable character, but the artist fell in love with her and made her the focus, which resulted in Rain Mikamura, the central female lead, becoming a generic Tsundere.
Now that it's clear how a Creator's Pet comes into action, what needs to be done before the general discussion becomes when the show Boned the Fish? Well, it's pretty complicated because once a Creator's Pet has been confirmed, the methods are difficult to pull off.
The second is developing the character, or at least admitting the faults they present. This works out better because the audience can finally see what is "wrong" with the Pet and, by making a conscious effort to work things out, the character winds up shedding the problems they presented and become a more universally approved entity. If the writers are able to avoid the trappings they fell into when the character was a Pet, the probability of the character becoming tolerable, even endearing to the audience can improve dramatically.
Another choice is to realize that you are stuck with the character, and instead of doing these solutions, it is better to focus on the other characters that exist and modify the story in which the Pet doesn't motivate the plot or advance it, but instead becomes part of the plot itself. Once that storyline is finished, you are free to do what you wish.
Finally, you can just keep the character as it is and ignore or annoy your fanbase. Why should you listen to them anyway?
As a final precaution, before you check and see about Creator's Pets, take a look at the fanbase itself. If the fanbase is historically difficult to please and represents the Broken Base in action, there might not be anything wrong with the character. Also, be aware of situations like the Vocal Minority, where who is griping might be a small group that doesn't share the general opinion. And, above all else, make sure that the character fits all the criteria set before this trope.