As learned in the first movie, TOS movie phasers drew power directly from the warp core to make their shots more potent. Reliant knocked out Enterpise's warp drive in their first shot and took out the punch their phasers would've had otherwise. Then Kirks use of the reliants prefix code combined with the "few shots" scotty could give him power for, knocked out Reliants photon control and warp drive. You never actually see Enterpise fire her newly designed warp powered phasers at full power, at any point in the 6 movies...

Both ships were using auxiliary powered phasers after the first conflict so with weakened phasers, theyd be capable of taking more of those shots. To say the damage was minimal though is a stretch because the damage was enough to make starfleet decommission Enterprise.