Sent Bitcore (BTX) to my Bitcoin Core wallet (a bitcoin wallet) - anyway to recover the BTX?
Private keys are independent of actually receiving any coins. You can get the private keys from your wallet by using the dumpprivkey or dumpwallet commands.

Alternatively, because Bitcore's wallet is based on Bitcoin Core, you can just copy and paste your Bitcoin Core wallet.dat file into the Bitcore wallet's datadir, replacing it's wallet.dat file (assuming that you are using bitcore's fork of Bitcoin Core).