= Solution
For the <curve generation criterion for an abelian variety>, first suppose $Y=X$ and an irreducible <Weil divisor> $D$ avoids $C$. The preceding part gives $K(\mathcal O_X(D))=X$, so $\phi_{\mathcal O_X(D)}=0$ and $\mathcal O_X(D)\in\operatorname{Pic}^0(X)$. A nonzero <effective divisor> cannot have this property: if $H$ is an <ample line bundle> and $g=\dim X$, then the <intersection product of Cartier divisors> $D\cdot H^{g-1}$ is positive, whereas an algebraically trivial <line bundle> has zero intersection with every curve. The latter follows from constancy of degree along a connected family defining algebraic equivalence; the former is the positive projective degree of $D$ after replacing $H$ by a very ample power. This contradiction proves that $C$ meets every irreducible <Weil divisor>.
Conversely suppose $Y\ne X$. Fix $c_0\in C$, so $C\subseteq c_0+Y$. Use the permitted fiber theorem to choose a <morphism of varieties> $f:X\to Z$ with $f^{-1}(z_0)=Y$. This morphism is nonconstant because $Y$ is proper. Choose an affine neighborhood $U$ of $z_0$. The irreducible image $f(X)$ has positive dimension, so its intersection with $U$ does too; some regular function $h$ on $U$ is nonconstant on this intersection. Then $F=h\circ f$ is a nonconstant rational function on $X$, regular on $f^{-1}(U)$.
The <pole divisor avoiding a fiber> construction applies: its pole <Weil divisor> is nonzero. Indeed on the smooth, hence normal, <projective variety> $X$, a rational function with no codimension-one poles extends to a global regular function; every global regular function on a connected <projective variety> is constant. Every pole component $D_0$ lies outside $f^{-1}(U)$ and hence avoids $Y$. Therefore $D_0+c_0$ is an irreducible <Weil divisor> disjoint from $c_0+Y$, and in particular from $C$. This proves the converse using precisely the fiber fact allowed in the question, without requiring a projective target $Z$. \b[The criterion is]
$$
\boxed{Y=X\quad\Longleftrightarrow\quad C\text{ meets every irreducible divisor of }X.}
$$
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