Forging a valid tag for an arbitrary message when you have some knowledge about the key I am using  HMAC-SHA256 with a 128-bit key. Assume my key generator leaks some information to an adversary. Hence, the adversary can predict the key with probability $\alpha$. Can I say that the adversary can forge a valid (message,tag) pair with probability $\alpha$?
[Suppose] the adversary can predict the key with probability $\alpha$.  Can I say the adversary can forge a valid (message,tag) pair with probability $\alpha$?


It's not quite $\alpha$.  Write it out with the chain rule:

\begin{align*}
  \Pr[\text{forgery}]
  &amp;= \Pr[\text{forgery} \mathbin\&amp; \text{guessed key}]
     + \Pr[\text{forgery} \mathbin\&amp; \text{didn't guess key}] \\
  &amp;= \Pr[\text{forgery} \mid \text{guessed key}] \Pr[\text{guessed key}] \\
  &amp;\qquad + \Pr[\text{forgery} \mid \text{didn't guess key}]
              \Pr[\text{didn't guess key}].
\end{align*}

Under your premise, $\Pr[\text{guessed key}] = \alpha$, so $\Pr[\text{didn't guess key}] = 1 - \alpha$.

Exercise: Figure out what $\Pr[\text{forgery} \mid \text{guessed key}]$ and $\Pr[\text{forgery} \mid \text{didn't guess key}]$ are.  Hint: Model HMAC-SHA256 as a uniform random function.  What's the probability any two prescribed inputs $m \ne m'$ collide under a uniform random function?  (Then add a term at the end for the distinguishing advantage of the adversary against HMAC-SHA256.)