Solder wick will work and is the best thing to use. However it sounds like you are not using it correctly. First of all it needs to be at least 5mm wide. Next put it down on the bench and put your iron over the the end. Then holding the spool move the wick through 180 degrees so it wraps round the tip. Then holding the tip and wick in tension move it and apply it to your joint ( mess ).
It also looks like your iron is not hot enough which will reduce the effectiveness of the solder wick as well.