What is the total impulse response in a system with feedback interconnection?
It is easier to work in the $s$-domain:

$$Z=YH_2$$
$$Y=(X-Z)H_1$$
Hence,
$$Y=(X-YH_2)H_1=XH_1-YH_1H_2\Rightarrow Y(1+H_1H_2)=XH_1$$
Therefore,
$$H(s)=\frac{Y(s)}{X(s)}=\frac{H_1(s)}{1+H_1(s)H_2(s)}$$
which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse Laplace transform.