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If this happens again you can get some isopropyl alcohol on strong cloth and gently rub around where the CPU contacts the cooler after undoing the clamping mechanism. This will help melt away and excess paste. You can also try gently pushing a plastic prying tool into the gap with some of the same alcohol on the end.
Isopropyl alcohol is my go to for removing paste.
Did that after keeping it stuck to the heatsink whilst straightening the pins as it provided a good solid base. Problem was that I didn't realise it had damaged the socket coming out since it wasn't unclipped at the time as you'd not know with the weight of the cooler and the level of resistance when lifting it out so RIP motherboard.