= Solution
For nonnegative integers $m,n$, put $B(z)=z^m[(z-a)/(1-\bar a z)]^n$. Since $|a|<1$, the denominator has no zero on or inside the unit circle. On $|z|=1$,
$$
|z-a|^2=1-z\bar a-\bar z a+|a|^2=|1-\bar a z|^2,
$$
so $|B(z)|=1$. Consequently $|-a|<|B(z)|$, and <Rouché's theorem> gives the same interior zero count for $B-a$ as for $B$. The latter has $m$ zeros at zero and $n$ at $a$, with the multiplicities combining when $a=0$. Therefore
$$
\boxed{B(z)-a\text{ has }m+n\text{ zeros in }|z|<1\text{, counting multiplicity}}.
$$
This is the <interior zero count for a finite Blaschke product minus a constant>. If $m+n=0$, the nonzero constant $1-a$ has zero zeros, so the same conclusion includes that endpoint case.
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