= Solution
Suppose that $f$ is not <surjective function>[surjective], and choose $y\notin f(Y)$. If $y$ lies in the open two-cell, then $Y\setminus\{y\}$ deformation retracts onto the target circle of the degree-$n$ attaching map, so its first <integral homology> is $\mathbb Z$. If $y\in Z$, the punctured space deformation retracts onto a finite graph and again has free abelian first homology. The factorization
$$
Y\xrightarrow{f}Y\setminus\{y\}\hookrightarrow Y
$$
therefore makes $f_*:H_1(Y)\to H_1(Y)$ factor through a free abelian group. Every homomorphism from the finite group $H_1(Y)\cong\mathbb Z/n$ to a free abelian group is zero, contradicting the assumed surjectivity of $f_*$.
The converse is false. The radial coordinate descends to a surjection $r:Y\to[0,1]$. The finite CW complex $Y$ is a <Peano continuum>, so the <Hahn-Mazurkiewicz theorem> supplies a continuous surjection $g:[0,1]\to Y$. Then $g\circ r:Y\to Y$ is surjective, but its induced map on $H_1$ is zero because it factors through the contractible interval.
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