The best advice I could give would be to take your time with your turns and trust in the probability that there is a solution for each turn. I don't know if it's an AI trick or something, but even on Hard difficulty, there are very, very few turns where you logically can't avoid losing a building. It's just a matter of finding it, similar to those old "checkmate in 3" puzzles in the newspapers.
Also a basic tip: a good way to start a turn is to identify the "critical" enemies: the ones that threaten the buildings. The rest can often be ignored (not really, but we're talking basic). Most of the time you'll have 3 mech actions, and 3 critical enemies, and you just have to tie those together. Later on, you will find yourself in situations where there are more than 3 critical enemies, and that's where you'll have to pull the real shenanigans.