One-time pad with public key changes
As @kelalaka pointed out OTP is not “differentially secure”. In addition it has the property, like all additive ciphers which work by modular addition of a keystream, that the keystream and the plaintext have symmetric roles.

What I mean is your suggestion to add the string $b$ to the key $k$ is essentially equivalent to the weakness resulting from reusing plaintext as in VENONA. 

The constant $b$ is visible to differential attacks, such as crib dragging. In this case, it is the keystream that is differentially deficient.