Lawn aeration question

Aerated my lawn this morning with one of those manual Yard Butler core aerators. Small yard, so wasn't worth renting one. My question relates to what to do with the plugs. I know you are supposed to leave them to decompose, but my home is only 2 years old, with EXTREMELY compact clay soil. The plugs seem to be essentially all clay. I can't imagine it being too beneficial to leave them in the yard. But I don't know shite, which is why I'm here. I've read conflicting info online on what to do with clay plugs. I'm located in Covington fwiw. Thanks