The clover you mentioned is probably white clover. Due to its growth close to the ground and the fact that its' shoots can root it is indeed a weed which is hard to handle. Aerating is one thing you can do, the other thing is herbicides. You will find the right remedy in your Garden Centre. But either way, watch the clover seeds! The seeds on the ground will not be harmed by aeration nor by the herbicides. So better mow them off before they are in flower.
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