Does XORing a random block into a layer of CBC encrypted message add some bit strenght? Let's suppose I encrypt a message with a block cipher in CBC mode, keep the key and IV secret and XOR a random block of data in each encrypted CBC block, being the XORed block the same for each CBC block (constant).
Does it add some encryption strength?
(I know this question is weird, but I need to know).
In general: No.
You have to assume the attacker knows what you are doing. That's Kerckhoffs' principle.
If your random blocks are actually a real One-Time-Pad on top of the block cipher, then it would be stronger - but you could achieve the same without the block cipher. If all your blocks use the same pad, this is true if the length is 1 block,  the pad is truly random, only used once and communicated using another secure channel.
If you just mesh different things together to make it more complex - that does not add security at all.
As a side note: The IV in CBC is not secret. If you consider it secret, you need to clarify how the sender and receiver both know it. And following that - do you consider that secret as static and reuse it? That would violate the requirements of an IV.