Any way to rewrite/simplify $\int dp ~f(p)~\delta[c-g(p)-E(p)]$?

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Im thinking of using $$\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}$$ but in my case $$\int dp ~f(p)~\delta[c-g(p)-E(p)]$$ the functions $f$ and $E$ are basically unknown, while $g$ is a simple square root function and $c$ a constant. Simplifying