Jensen's gap between $| \int f \,d \mu | ^2 $ and $ \int |f|^2,d \mu$ especially when $\mu = \mathbb{P}$

Question

Do we have tools by which we can measure $$ \int |f|^2 \,d \mu - \left| \int f \,d \mu \right| ^2 $$ especially when $\mu = \mathbb{P}$?

I am especially interested in the cases when $f$ is such simple that $f(x) = x + c$ with some constant $c$ and $x \in \mathbb{R}$, and with an integral domain $(a, \infty)$:

$$ \int_a^\infty (x+c)^2p(x)\,dx - \left|\int_a^\infty (x+c)p(x)\,dx \right|^2 $$