The Atoner: To some extent; his grandfather was ashamed at working on nuclear weapons for the Eastern Coalition that killed millions in World War III and dedicated the rest of his life to working on peaceful projects like the warp five engine. Although Hwai doesn't feel personal guilt for his grandfather's actions, he does consider his work to be a continuation of that atonement.
Badass Boast: "COME AND HAVE A GO IF YOU THINK YOU'RE HARD ENOUGH!"
Character Development: He gets more open-minded about aliens as the series goes on, his early experience of them mostly having been with arrogant Vulcan ambassadors on Earth or unpleasant criminals he fought.
The Unintelligible: Not entirely. As written, he can be understood, assuming you understand thick North Yorkshire dialects. If you don't, he's this trope. Often treated like this in-universe by the crew (partly jokingly, though).
Legacy Character: Inverted - he is the ancestor of the Lieutenant Stiles from the Star Trek: The Original Series episode "Balance of Terror", who mentioned in that episode that several of his relatives had served (and some had died) in the Romulan War.
I Coulda Been a Contender: Initially has this attitude to the Hwai and the Enterprise crew over them getting all the glory in their exploration missions and the follow-ups made by ships like the Dauntless being far less well known.
Military Maverick: Certainly compared to other Romulan officers, in that he actually considers his enemies to be intelligent beings (albeit barbarians) rather than animalistic. He is also willing to be 'creative' with orders from above when necessary, though he won't refuse a direct order from the Praetor.
My Country, Right or Wrong: In contrast to most of the Romulans, who wouldn't even stop to consider that there could be a 'wrong'.
Karma Houdini: Has her command reinstated, albeit with a lower profile, after the disaster of Rigel (though it wasn't entirely her fault) rather than being punished. Others wonder why this is, not knowing of her Tal Shiar connexion.
Named After Somebody Famous: Inverted - it's implied that the IRW Haakona from the Star Trek: The Next Generation episode "Contagion" is named after her. Lampshaded when the Subcommander tells her that the Tal Shiar can ensure that she so escapes disgrace that she will have ships named after her in the future.
We Are Struggling Together: Hampered by internal political tensions among her subordinates, ultimately resulting in a coup when her original plan fails.
Klingon Promotion: Takes Haakona's command via a mutiny, albeit not one that includes killing her.
The Mark: But he doesn't realise it till it's too late.
Oh Crap: He is intelligent enough to eventually realise what a disaster his Attack! Attack! Attack! tactics - the product of his arrogant assumption that all non-Romulans are cowards who will easily be intimidated into surrender - have landed the fleet in. Too late.
Skewed Priorities: Like the rest of the Tal Shiar, he thinks throwing away a task force as part of a ploy to take down one political opponent in the middle of a war is an OK deal. Helps that, like most Romulans, he's so confident the war will be won easily that it doesn't matter.