Does the ability to influence the key of a MAC effects its security? If a PPT adversary can influence the key of a MAC function, is it still secure?

For example, if we define $f$ as follow:

$f(r,x) = HMAC_{(k\oplus r)}$(x)

If the adversary has oracle access to $f$, how likely he can predict the key $k$?  

For respecting the persons who have answered the above question, I do not change it. The actual scenario is as follows:  

I have a module that gets $(m,r)$ and generates $(m,x,HMAC_{k \oplus r}(m \parallel x))$ where the $k$ is the secret of the module and $x$ is the message added by the module. So:  

$f(m,r)=(m,x,HMAC_{k \oplus r}(m \parallel x))$  

Can I claim that every tuple $(m,x,t)$ s.t. $t=HMAC_{k \oplus r}(m \parallel x))$ is generated by the module? 

P.S. I am still curious about the above claim. But, I just noticed that if I change $f$ as follows:

$f(m,r)=(m,x,HMAC_k(m \parallel x \parallel r))$    

then, I can claim every tuple $(m,x,t)$ s.t. $t=HMAC_k(m \parallel x \parallel r))$ is generated by the according to the definition of MAC.
If a PPT adversary can influence the key of a MAC function, is it still secure?


In general probably no if you make no restrictions on the ways the key can be influenced. In particular the answer is "no" when you rely on your MAC actually being a PRF for your security proof (like HMAC does). This means that if your MAC relies on this then you need some additional argument for related-key security, as PRFs cannot be secure against all classes of related-keys. For a proof and more background on this see Proposition 8.4 of the original paper that introduced this (PDF).


  If the adversary has oracle access to $f$, how likely he can predict the key $k$? 


Cryptography isn't about "can you predict the key" because you can usually build schemes that don't use their key-input and are thereby trivially secure if all you want is the key.

A more interesting question would be, if an attacker has access to (this concrete) $f$, how easy is it to forge tags? This has in fact been studied for HMAC and keys related by a constant leading to an attack that runs in time $2^{n/2}$ with $n$ being the HMAC's output length.