Can I play 1 energy to give Renekton, Brute 1 might on the enemy turn?

No. Renekton, Brute's ability [1]: Give me +1 [M] this turn is a standard activated ability with no Reaction tag, so it can only be activated on your own turn, during your Main Phase, in an Open State (Rules 145.2, 381). On your opponent's turn you have no such window for it.

The only thing that can fire on the enemy turn is his triggered ability ("When my Might becomes 10 or more, empower me") — but you can't manually pay the 1 Energy to make that happen off-turn.