I don't think either of the comments under the question quite get at what's going on here.
My understanding of it is this (and please correct me if I am wrong): a specific value for $p$ is drawn from a known distribution, but we don't observe that $p$. Instead we observe a binomial random variable with that $p$. You want to use both the information in the known distribution for $p$ as well as in the likelihood to make some inference about $p$.
On that understanding:
A random $p$ obtained from a known distribution plus some information about that specific draw of $p$ from a sample is naturally answered with Bayesian approach; the known distribution of $p$ is the prior, which you then update via the sample (specifically the posterior is proportional to the product of the likelihood and the prior).
For example, if the prior were beta, the posterior would also be beta (since the
beta is conjugate for a binomial likelihood).
If the prior were symmetric triangular, the posterior would be piecewise beta (with two continuous pieces).
More generally we could use a variety of techniques to obtain the updated information about $p$. On this simple problem - assuming a prior that didn't work nicely with the likelihood - one option would use numerical integration to scale the product of the prior and the likelihood to a density.