It will not be strictly necessary to remove the old solder; just adding new solder in order to get a good welding should be enough. (but first of all you must clean the remainders of melted solder around, presumably produced by the electrical sparks due to the bad contact).
The foto looks like there is only a bad welding, so it should be no necessary to redo the other three... I will not redo the other three, but you can try if you feel confident and want to be sure.
Soldering electrical componentes is not such a difficult task. The "secret" is to apply heat enough to the parts to be connected and not only to the solder (but no to the point of damaging the components and/or the pcb !!).
For this concrete task:
- use a 25W soldering iron (like a JBC model 30 ST or similar) with a tip of 2 or 3 mm diameter,
- plug the soldering iron, wait enough time for the soldering iron to reach the working temperature,
- clean the tip with a wet sponge, apply just a bit of solder to the tip, (the tip should look now clean and brilliant with the solder closely attached to it, without forming a kinf of "ball" due to excess of solder), if required use the wet sponge again to retire the excess of solder away from the tip,
- apply the soldering iron tip to the coil souspicious welding, wait 1 or 2 seconds until the solder melt, be sure to reach the coil terminal with the tip of the soldering iron to assure that you are applying heat to the coil terminal also,
- add now new solder by approaching the solder wire to the tip of the soldering iron and melted solder; be sure the new and old solder (if not previously removed, as you said) are completely melted together, wait one second, take out from the solder the soldering iron tip... and that's all.
The solder should flow freely; if it’s forming balls then there is likely a problem with the tip, the temperature, or (sometimes) the surface. A finished good welding must look brilliant and take a form similar to a hill, not like a ball (something like this "^" and not like this "o"). The complete task should not take more than 5 or 6 seconds.
Do NOT use lead-free solder. You should use lead-free ONLY if you are experienced using it and confident. Lead-free solder requires a higher
melting temperature and thus is more difficult to use. I will not recommend lead-free solder for first-time practitioners.
I appreciate very much your offering but don't worry about that... I hope you will be lucky and that the only problem with the board is a bad contact in the coil.