When you hit a structure with your pickaxe you will see a blue circle appear. This is the weak point. If you position the aiming cursor over the blue circle on the next swing you will hit the weak point and do double damage.

In order to complete this quest you need to hit 20,000 weak points. This could come with natural progression if you play enough. If you really want to complete this quest you should prioritise structures with high health too make sure you can hit as many weak points as you can.

Either way this is a quest I wouldn't recommend investing time in if you have other quests available as going for them may be a better use of your time.