= Solution
Treat $Q$ and $B$ as mean detected counts, or as photon counts with unit <quantum efficiency>. Let the independent patch measurements be $X\sim\operatorname{Poisson}(Q+B)$ and $Y\sim\operatorname{Poisson}(fB)$. For the specified unweighted <background subtraction> $\widehat Q=X-Y$,
$$
\mathbb E\widehat Q=Q+(1-f)B,\quad
b=\mathbb E\widehat Q-Q=(1-f)B,\quad
\operatorname{Var}\widehat Q=Q+(1+f)B.
$$
The first error is a fixed <bias of an estimator>; the last expression is the sum of independent <photon shot noise> variances. Since $Q,B$ are already patch totals, no extra factor of the pixel count $n$ is needed, and <read noise> is neglected.
To obtain the systematic-error ceiling requested in the following clause, define the accuracy measure $Z$ using total root-mean-square error relative to the true source count. By the <bias-variance decomposition of mean squared error>,
$$
\boxed{Z=\frac{Q}{\sqrt{\mathbb E[(\widehat Q-Q)^2]}}
=\frac{Q}{\sqrt{Q+(1+f)B+(1-f)^2B^2}}.}
$$
For $f\simeq1$, the shot-noise term is approximately $Q+2B$, but the mismatch term must be retained.
There is a terminology qualification: the usual variance-based <signal-to-noise ratio in photon counting> is $Q/\sqrt{Q+(1+f)B}$ and does not include a fixed bias as noise. The printed next-part limit requires the root-mean-square accuracy convention above. A deterministic background mismatch contributes $b^2$ to <mean squared error>, not to the statistical variance.
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