= Solution
The <Duhamel principle> superposes the response to source impulses:
$$
\boxed{\Psi(t,x,y)=\int_0^t
\frac{p(s)}{4\pi D(t-s)}
\exp\left[-\frac{x^2+y^2}{4D(t-s)}\right]ds.}
$$
This expression uses the source at the origin, is causal, and has zero initial data in the distributional sense. Its integral over the plane is $\int_0^tp(s)\,ds$, consistent with the injected total amount.
For the finite-duration source and $t>t_0$, set $b=(x^2+y^2)/(4D)$. The <causal diffusion from a finite-duration planar point source> becomes
$$
\Psi=\frac{p_0}{4\pi D}\int_{t-t_0}^t\frac{e^{-b/\tau}}{\tau}\,d\tau.
$$
With $\eta=b/\tau$, or by differentiating the supplied <exponential integral>, this is
$$
\boxed{\Psi=\frac{p_0}{4\pi D}
\left[\operatorname{Ei}\left(-\frac b{t-t_0}\right)
-\operatorname{Ei}\left(-\frac bt\right)\right].}
$$
If $b/(t-t_0)\ll1$, both arguments are small. Their logarithmic constants and $\ln b$ cancel, leaving
$$
\boxed{\Psi(t,x,y)\approx\frac{p_0}{4\pi D}\ln\frac{t}{t-t_0}.}
$$
At the origin this logarithmic expression is exact for $t>t_0$, obtained as the limit $b\to0$. The leading spatial correction is $-p_0b\,t_0/[4\pi D\,t(t-t_0)]$. If also $t\gg t_0$, the leading value is $p_0t_0/(4\pi Dt)$.
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