Fan cools solder down too quickly
No, you don't need a special extractor. And there is a way to work around the cooling effect of the fan.

It's called a decent soldering iron, and you need one desperately. This is not due to the specific problem which led you to post, although that's bad enough. It's taking a long time to melt your solder, and during this whole time your parts are getting heated. This is very bad for your parts. You can easily kill ICs by doing this. Bite the bullet and get a decent iron.

It's also possible (but you don't mention it) that your cheap kit gave you solder which is too thick. For general electronic soldering, .050 solder is about as thick as you should have, and .025 dia solder is much better.