Question about ε-moves at $\varepsilon NFA$
In an $\varepsilon NFA$, when processing an $\varepsilon$ character, we can move on an $\varepsilon$-edge or we can stay on the state we are currently in. 
And for example since $a= \varepsilon a \varepsilon\varepsilon$, in your $NFA$ these two paths are both correct:

$$p \rightarrow^{\varepsilon} p \rightarrow^{a} q \rightarrow^{\varepsilon} s \rightarrow^{\varepsilon} s$$

and

$$p \rightarrow^{\varepsilon} p \rightarrow^{a} q \rightarrow^{\varepsilon} q \rightarrow^{\varepsilon} s$$

Therefore, with the character $a$ we have $\delta(p,a)=q$ and for the word $a$ we have $\hat{\delta}(p,a)=\{q,s\}.$