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A smooth projective curve $C\subset A$ generates $A$ by its point differences if and only if it meets every prime Weil divisor. If a prime Weil divisor avoids $C$, its restriction to $C$ has degree zero; constancy of line bundle degree in a family shows that every translate either contains $C$ or misses it. This forces invariance under $C-C$ and puts the generated subgroup inside the line bundle translation stabilizer. If this subgroup were all of $A$, the divisor would be algebraically trivial, contradicting its positive intersection with an ample line bundle. Conversely a proper closed generated subgroup admits a disjoint divisor by the pole divisor avoiding a fiber construction, after translating the fiber to contain $C$.

Pull $L$ back by the morphism of varietiesThe fiber of this line bundle over $x$ is ${\phi }^{\ast }{T}_x^{\ast }L$. An abelian variety is connected, so the preceding constancy of line bundle degree in a family applies. Consequently $\boxed{\mathrm{deg}({\phi }^{\ast }{T}_x^{\ast }L)=\mathrm{deg}({\phi }^{\ast }L)}$ for every $x\in X$.