a quick language question

Mewtwo will have a marked influence on all aspects of battle, but it will not trump any- and everything.

In this sentence, it means that Mewtwo is almost (but not really) metagame-breaking. It makes sense if you imagine the metagame as a table that Mewtwo is sitting on.

(A metagame breaker is an option - in this case, attacker - that is too powerful and makes many other options useless.)

"Breaking a game" is usually figuratively referring to an object or mechanic that is so overpowered that it ruins any balance from costs or counters. (Something "game breaking" can also more literally be something that stops a game from working all together.)

By context, "weighs heavily upon it" is probably meant to say that it pushes the limits, but not far enough to ruin anything too badly.

I'm guessing this is in the context of Mewtwo's base stat nerf. If Mewtwo had been released with its previous base stat calculations, it would have been the top counter for just about every fight. After the base stats were cut by 9%, it is still one of the best counters when it has a type advantage, but falls a little lower on the list of top attackers when it doesn't have a type advantage.

Breaks the game: Mewtwo is so powerful that no other pokemon is worth mentioning any more because Mewtwo is better in all cases.

Weighs heavily upon the game: Mewtwo should be considered in nearly every matchup as a viable and possibly superior option regardless of type advantage. Having access to Mewtwo(s) allows great flexibility in all aspects of battling.

Mewtwo isn't Bo Jackson Tecmobowl.