The Criminal Injuries Compensation Authority’s minimum award is £1,000, and you can only apply (or at least, will only be successful) if your injuries are serious enough to warrant such an amount.
Looking at Magistrates’ guidelines which help them assess compensation, a black eye warrants up to £125 and a cut lip between £100 and £500 (though it is doubtful that both sums would be awarded in full). There is no guidance for hearing loss and, unless it was permanent or at least long lasting, I suggest compensation would be minimal.
In short, it does not sound as if the injuries suffered by your partner’s cousin are substantial enough to be considered by CICA. He may, though, be eligible if a conviction is secured against his assailant.