Psi units can dominate one unit per mission. Use this ability to dominate a powerful unit, such as an Andromedon, or a gatekeeper if you have encountered this unit. This serves two purposes:
- You can use this unit to deal more damage, or attack more units
- You can push this unit forward to tank some damage, preventing that damage from being dealt to your own units
Psi units also have access to the null lance, a powerful psi attack that deals damage to a small straight line. Psi units can also open a rift in a circle that deals damage, and will make some of the units panic (deals more damage too) and some of them mind-controlled (will deal damage too). The mind control will not last long, but will make the enemy units focus on taking out that unit. Isn't it wonderful?
You can set up a sniper to take an overwatch shot against units that move in a certain cone of fire. This allows the sniper to shoot 3 enemies. The serial ability allows the sniper to fire multiple shots as long as those shots kill the enemy. This too allows the sniper to kill three enemies if set up properly.
Set up your rangers with the untouchable perk and the perk that allows you to move the unit after a kill. The first shot against this unit in the enemy turn will deal no damage. Use the ranger to flank an enemy, murder it, then move the unit to a place where it will likely only be shot once. You can use the rapid fire ability to more or less guarantee a kill.
Heavy units can fire a grenade and then shoot if they have the salvo perk. If they have the holotargeting perk it makes it more likely for your other units to kill the units they target. Heavy weapons remove a lot of armour, making other attacks also more useful. Finally, if you equip them with the W.A.R. suit, you can use plasma cannons to deal a lot of damage in a straight line, or shredder cannons to deal a lot of damage in a cone of fire, while also removing armour.
Archons have three attacks. They have a melee attack, which is painful. They can shoot a plasma bolt at an unit, which is painful. And they can use blazing pylons, which they are likely to use if they can target multiple units with it.
Codexes can teleport, which will not trigger overwatch of units, and have two attacks. They can fire their gun, or they can open a rift over some units which will disable their primary weapon, but deals no damage whatsoever. They have the annoying habit of multiplying though when damaged, so you should try to one-shot them, or leave them alone until you can deal with them.
Heavy robots can shoot your units, use melee or they can use a powerful grenade attack.
In general though, you likely can not kill them all in one turn. Your assessment that "you need the grenades in the last fight" is also somewhat void. You need to use grenades whenever there is a group of enemies that you need to deal damage to, or need to remove armour of. When you realize that you might get overwhelmed, you'll need to use distance to your advantage.
Focus on getting the number of enemies down. Look at your units and determine how much damage you can likely do. Targets that are closer to you are more dangerous. Targets that need to do a double move to do damage to you can wait until the next round. Move your units in such a way that you can deal damage to the units you want to deal damage to, and other units that might want to attack your unit have to take low-chance shots or have to double-move to get into better positions.
In your case I would probably try to get the heavy MEC's to low health with a shredder cannon, then use a void rift to finish most of them off. Alternatively you can use a ranger and some sniper fire. Focus on the archons next. One of the is probably going to use blazing pylons, which you can move away from. The second one probably tries to do damage, either by shooting/melee or by going into overwatch.
If too many enemies are in range, do a full retreat. Try to get the units into better positions and let the enemy move towards you. They either cluster up, and you can use AOE to do a lot of damage, or they come in waves, which you will have a better chance of killing. You usually should avoid triggering this many units to attack at once.
Good luck, commander.