"Mag-away" is "to quarrel" and it means fighting between 2 or more individuals or groups, usually on a personal level and often over petty things. This would include physical acts like shouting, insulting each other, hair-pulling, scratching, biting, etc.
Mag-away Example Sentences in Tagalog: (4)
Angdalawángbatangnag-aaway ay ayawmagpasawáysakaibigannilá.
The two kids fighting would not heed the admonishment/urging of their friend to stop.
The couple is quarreling.
It won't take long before those two kids will quarrel.