If the plaintext contains the key used to encrypt the ciphertext, is the ciphertext easier to decrypt?
There aren't any known generic attacks against this setup to work on arbitrary ciphers. On the other hand, the standard notion of IND-CPA security does not imply security when encrypting your own key.

I don't know of any commonly used block or stream cipher which has trouble with this setup, but one could be created and might even exist.