How do you improve relations with the Hai?

[Hacker]

How do you improve relations with the Hai? And all of the other factions for that matter

[warp_core]

Repair their disabled ships. This only works when you have a positive reputation with them though so if they don't like you, it's not possible.
You can attack their enemies. These can be found in governments.txt in the game data. Beneath each government entry will be a set of attitudes towards other governments. The governments that have an attitude toward them of more than 0.05 or less than -0.05 are governents where an action towards them will affect the first government.
Say, for example, Hai have an attitude towards Merchants of 0.1. This means that doing something to a Merchant government ship that affects your reputation with that government will result in a change to your reputation with the Hai of 0.1 times that change.
The Hai have an attitude toward the Unfettered of -0.5 so doing things to Unfettered ships that lower your reputation with them will raise it with the Hai. Things that do this are: disabling an Unfettered ship; boarding an Unfettered ship (when hostile to them); capturing an Unfettered ship; destroying an Unfettered ship.
So basically, to raise your reputation with the Hai, repair Hai ships and disable, capture, plunder and destroy Unfettered ships.

A spoiler tag in the title may also be necessary.

[Avior]

Be careful about topics like the Hai. Use [spoiler] tags when necessary. (When in doubt, use them.)

For the most part, you can repair their disabled ships in combat, which will give you a boost of c * k * 0.1 reputation, where c is the boarded ship's crew complement (in 0.9.9, you'll be able to check their crew with the Tactical Scanner outfit); and k is equal to 0.5.
In the case of the Hai, they are hostile to the Unfettered, and attacking the Unfettered does indeed affect your reputation with the Hai: a boost to your reputation of (source)[github.com]

• c * k * 0.5 for disabling;
  
• c * k * 0.3 for boarding;
  
• c * k * 1.0 for capturing;
  
• c * k * 1.0 for destroying.

But note that not all species will behave this way. Some may be hostile to another government, but attacking that government will not actually affect your reputation. You can check this in https://github.com/endless-sky/endless-sky/blob/master/data/governments.txt: if an "attitude toward" line is in the range -0.05 < k < 0.05, you cannot change your reputation by attacking that government. In addition, the impact of an event (based on the list above) will vary depending on the attitude toward value, which in the case of Hai versus Unfettered is k = 0.5.