Python 2, 84 81 bytes

lambda c,a:reduce(lambda t,b:all(b)and(0,0)or((t[0]+b[0])%24,(t[1]+b[1])%60),a,c)

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EDIT: Saved 3 bytes by replacing b[0]&b[1] with all(b); thx 2 Eric The Outgolfer.

Function that takes time as a tuple (hour,minute); outputs same way.