Is that a big relay? Looks like a standard size to me.
I have used a solder sucker to take a lot of the solder away and then remove the rest with braid. If you do not have braid, then the outer screen of co-ax cable can be used. The thinner the wire the better so that capilarity attraction is high. The braid/cable must be clean and a bit of flux can be a help.
If there is a little bit of clearance, the pins can be moved to the centre of the hole so the solder does not grab again.
I now have a heated solder sucker, this is like a soldering iron but with a hollow bit. Put it over the pin, wait for the solder to melt and then push the button. It clears the solder down the hole quite well but you may have to centre the pin.
For integrated circuits I have made a wide bit for my iron using a piece of copper central heating pipe, squashed in a vice so that it jams on the end of my iron. I can do eight at a blow!
Some people use hot air blowers but I have never tried this.