  In this paper, we mainly study the regularity of pullback
$\mathcal{D}$-attractors for a nonautonomous nonclassical diffusion equation
with delay term $b(t,u_t)$ which contains some hereditary characteristics.
Under a critical nonlinearity $f$, a time-dependent force $g(t,x)$ with
exponential growth and a delayed force term $b(t,u_t)$, we prove that there
exists a pullback $\mathcal{D}$-attractor $\mathcal{A}=\{A(t):t \in
\mathbb{R}\}$ in $\mathbb{K}^1=H_0^1(\Omega) \times L^2((-h,0);L^2(\Omega))$ to
problem \eqref{ine01} and for each $t \in \mathbb{R}$, $A(t)$ is bounded in
$\mathbb{K}^2=H^2(\Omega) \cap H_0^1(\Omega) \times L^2((-h,0);L^2(\Omega))$.
