Is there a generalization of information theory to polynomially knowable information?
Yes. Time-bounded Kolmogorov complexity is at least one such "generalization" (though strictly speaking it's not a generalization, but a related concept). Fix a universal Turing machine $U$. The $t(n)$-time-bounded Kolmogorov complexity of a string $x$ given a string $y$ (relative to $U$), denoted $K^t_U(x | y)$ (the subscript $U$ is often supressed) is defined as the shortest string $p$ (a "program" for $U$) such that $U(p,y)=x$ and such that the computation of $U(p,y)$ takes at most $t(|x|)$ time. If you take this as your definition of "conditional information", then you can likewise define all the usual concepts from information theory. 

However, in this time-bounded setting, not all of the usual theorems of information theory are known to hold. For example, symmetry of information is known to hold for usual Kolmogorov complexity (no time bound), but not known to hold for time-bounded. See, for example, Chapter 6 of Troy Lee's thesis.

If you are concerned that this applies to strings rather than distributions, I suggest reading the following papers, which say that in fact Kolmogorov complexity of strings and Shannon entropy of distributions are very closely related:


Grunwald and Vitanyi. Shannon Information and Kolmogorov Complexity
Hammer, Romashchenko, Shen, Vereshchagin. Inequalities for Shannon Entropy and Kolmogorov Complexity.


(On the other hand, there are some properties that are known not to be shared between the two, see Muchnik &amp; Vereshchagin, Shannon Entropy vs. Kolmogorov Complexity.)