Given $f$ is positive and continuous on $[a,\infty)$ and $\int_a^\infty f$ converges, does there exist $0<c<1$ such that for all $c\leq p\leq 1$, $\int_{a}^\infty f^p$ converges?
First of all I found this statement is true for functions of the kind $f(x)=\frac{1}{x^\alpha}$. Intuitively, taking the power of $p$ makes values of $f$ that are above 1 (which are still possible given $\int_a^\infty f$ converges) , decrease towards 1, and values of $f$ that are below 1, increase towards 1.
Following your idea of testing with functions $f_{\alpha} = x^{-\alpha}$, combined with the fact that taking power $p$ increases the function, note that $\int_a^{\infty} f$ converges for all $\alpha > 1$, but $\int_a^{\infty} f^p$ converges only when $\alpha p > 1$. Hence, $c > \frac 1 {\alpha}$. What can you conclude?