If you're looking for get-in/get-out type contracts, these are typical. There are probably others:
Recon raid -- Test an enemy force on Planet X. Get out.
Objective raid -- Destroy something on Planet X. Get out.
Blow through a few bugmechs on your way in/out? }:)
Extraction raid -- Capture something that can be carried off Planet X. Get out.
Hmm. I'll need at least one map that has a building, plus an allied troop transport and infantry to enter and acquire the package.
Diversionary raid -- Distract enemy force on Planet X from operation by allied forces. Get out.
Hmm. Attack an enemy base and run for it? Would probably need to stay on-planet (or make the enemies
believe so) long enough for the allied forces to achieve their objective, then link up at the JumpShip.
Relief Duty -- Resupply and/or temporarily reinforce allied forces already engaged on Planet X.
I'm seeing enemy 'Mechs facing off against allied militia vehicles. What kind of raider would enjoy having a lance of heavy and medium 'Mechs show up in their rear?
Bandit hunting -- Seek and destroy bandits on Planet X.
Possible variations include forcibly removing a merc unit that went rogue and is acting as local tyrants, or waiting in ambush for a group of bandits that have gotten too predictable for their own good.
Longer-term or more indefinite contracts that are driven more by the GM's NPC forces could include:
Garrison Duty -- Protect/defend something stationary on Planet Z.
During which time any of the shorter-term ideas could happen as an enemy force.
Security Duty -- Protect/defend something mobile (like a political leader or CEO) on Planet Z.
I'm reminded of the Peace Talk mission in MW4: Mercs that got bombed by Liao forces, and the subsequent mission to escort the survivors to the spaceport.
Riot Duty -- Control unruly civilian population on Planet Z.
Not a fan of this one for many reasons, the most practical of which is that the unit only has one 'Mech that carries a weapon system suited for anti-infantry/peasant-suppression duties.
Guerilla Duty -- Go to ground on Planet X and organize resistance to enemy forces.
Cadre Duty -- Train allied forces on Planet Z (usually in combination with Garrison Duty).
You could probably combo these two if the rebels have gotten access to 'Mechs of their own. Their 'Mechs would probably be mediocre at best, but that's better than having no 'Mechs at all.