4. Given real number $a \geqslant-2$, and
$$
\begin{array}{l}
A=\{x \mid-2 \leqslant x \leqslant a\}, \\
B=\{y \mid y=2 x+3, x \in A\}, \\
C=\left\{z \mid z=x^{2}, x \in A\right\} .
\end{array}
$$

If $C \subseteq B$, then the range of values for $a$ is
4. $\left[\frac{1}{2}, 3\right]$.

From the problem, we know $B=[-1,2 a+3]$.
To make $C \subseteq B$, it is only necessary that the maximum element of set $C$ is in set $B$.
$$
\text { Hence }\left\{\begin{array}{l}
(-2)^{2} \leqslant 2 a+3, \\
a^{2} \leqslant 2 a+3
\end{array} \Rightarrow \frac{1}{2} \leqslant a \leqslant 3\right. \text {. }
$$