For some reason the solder wick just wont absorb it and they don't stay molten enough for the sucker to suck it.
Is the sucker here a manual tin sucker?
If so then heat it from up and suck from down side.
Don't block the hole with the soldering iron.
Side here is not the board but how you place it.
You can also try tapping but it seems that for that those holes might be too small.
There you tap the board from down to up so that molten tin drops out.
The hole wont become empty but possibly clear enough for the connector.
Blowing is as equally possible as is sucking.
You just need to make a suitable nozzle, put there some side holes also for extra air.
Then just heat the hole from down and blow from up.
Sides are again how you place the board.
Key here is a full and truly molten hole.
When new tin drops in it's all molten.
Those desoldering needles may feel uncomfortable but you can use any material that can take the heat.
There you should place the board so that component side is up.
Then heat from down and push the stuff down from up.
Use wood if you're affraid that the stuff gets stuck, you have some hard ones there.
Possible leftovers will flatten when new connector is soldered in or you can cut them away.
Don't pull the stuff back, leave it for later when tin is hardened.
When you put the new connector in don't let it partially drop out.
On the other hand, you have a practice so...