Let $p$ be a prime integer and $\hat{\mathbb{Z}_p}$ be the $p$-adic completion of $\mathbb{Z}$. If $f(x)\in \mathbb{Q}[x]$, the polynomial ring over $\mathbb{Q}$, and $a\in\hat{\mathbb{Z}_p}$
I want to know what is the meaning of $f(a)\in p\hat{\mathbb{Z}_p}$ is. More precisely, I do not understand the meaning of $f(a)$ and I do not know what the set $p\hat{\mathbb{Z}_p}$ is.
As to your questions, if for example $f(x) = x^2+7x+1$, then $f(a) = a^2+7a+3$. Like, if $a=2$, then $f(2) = 2^2 +7\cdot 2 +3 = 21$.
And "$p\Bbb{Z}_p$" is $\{p\cdot x: x \in \mathbb{Z}_p\}$. Which just means all elements of $\Bbb{Z}_p$ which are multiples of $p$.
So for the above example $f(x) = x^2+7x+3$, if $a=2$ and $p=7$, then $f(a) \in p \Bbb{Z}_p$.