a multishot pcp would be the best for hunting, however can be very expensive.
for about 200-300 you could get a top quality used springer with a decent scope.
after plenty of practice you should have learnt how to shoot it properly and as long as its pushing out over 10 ft lbs will be fine for hunting if you can group ever shot consistently under a 2p sized target at your chosen range.
good guns to look out for cheap , up to £300 , would be
HW 95
air arms tx200
then maybe up to about £350 for a good condition
HW97 or 77
or aa pro sport