Binomial conditional on Poisson

I have a binomial $CNTsimmathcal{B}(n,p)$, and a poisson $Psim mathcal{P}(lambda)$.

Now I’m looking at $CNT$ conditional on $P$:

What’s the expected value, $E(B|P)$? What is the variance $Var(B|P)$? What is the covariance $Cov(B,P)$? Are there closed forms, or must we always go through the sums?

After feedback in the comments, here’s what I have. Going through the sums:

$$
E(textbf{CNT}|A^*) = sum_{w=0}^{w^*} sum_{v=-w}^{+w} vP(text{CNT}=v|A^*=w) P(A^*=w)
$$
$$
= sum_{w=0}^{w^*} sum_{k=0}^{w} (2k-w) P(text{CNT}=(2k-w)|A^*=w) P(A^*=w)
$$

$$
=sum_{w=0}^{w^*} sum_{k=0}^{w} (2k-w) binom{w}{k} p^{k} (1-p)^{w-k} P(A^*=w)
$$

where $w^* = 4lambda$, $text{CNT} sim{B}(w,p)$, and $A^*sim{P}(lambda)$.

This, and the corresponding for variance, yields correct numerical values. What I wonder is whether or not there are closed forms for $E(CNT|A^*)$, $Var(CNT|A^*)$, and $Cov(CNT,A^*)$.