In the ordinary Kelly criterion the game is that you get to place a bet $B$ and your new wealth would be:
$$W^+ = \begin{cases} W + cB & \text{ with the probability of }p\\ W - B & \text{ with the probability of }q = 1-p\\ \end{cases}$$
and the criterion is that the optimal bet size is $W(cp-q)/c$. Now consider that we include a fixed income $\phi$ per game, that is the new wealth would instead be:
$$W^+ = \begin{cases} W + cB + \phi & \text{ with the probability of }p\\ W - B + \phi & \text{ with the probability of }q = 1-p\\ \end{cases}$$
What then would be the optimal bet size?
My thought is that you shouldn't expect the situation to be as homogenous as in the classical example. What I'd expect is that given $W/\phi$ being constant it would be homogenous. I also expect it to converge to the classical result as $W/\phi\to\infty$. That is
$$B^* = {W(cp-q)\over c}{1\over \sigma(W/\phi)}$$
For some $\sigma$ such that $\lim\sigma = 1$.
In the proof on wikipedia of the classical situation one notes that the outcome after a number of wins and losses doesn't depend on the order. I tried assuming that, but it looks like this din't work out (I couldn't find any $\sigma$ having that property even if $c=1$).