Filling the hole with solder is not an issue - they are ‘through holes’ and designed to flow the solder along the pin poking through them.
I would work from the back as it looks like you have a short to the pin next to the hole. The join next to it may also be a bit dry - just heat with a clean iron and let it flow.
You may find that a flux pen will help - often the flux on lead free solder will evaporate so quickly it does not help the flow.
Put the wick on the solder - heat till you see the solder come through, remove iron, remove wick, move wick down and repeat as needed.
I’ve seen some engineers cut the wick after each ‘suck‘ to the remove the risk of it melting again.