I'm not a lawyer, so this isn't legal advice. For that, you do need a lawyer. However, you're generally not protected if you "misread" something in the contract. If someone altered the contract after you signed, that's a different matter. But you're responsible for reading it . . . and reading it correctly.
One thing that might, just might, work is that delay in your receiving the contract. In certain cases, you're supposed to be provided a copy immediately. If there was some sort of regulatory requirement that you be provided a copy of the contract and the other party failed to abide by that, you might (again, not legal advice) be able to get out of the contract. So, be sure to mention that to the lawyer.
Hope that helps.
But from what you describe I don't see any valid reason why you would be able to void this. I don't believe there is a timeline on getting your copy of the contract. If you had never gotten one, maybe things would be slightly different.
So what did you misread, if I may ask?