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I am trying to assemble the answers to the question(s) that were hashed out in the comments (and also in a separate discussion with Jonny Evans). This answer is community wiki since it is the outcome of collaborative discussion. Please feel free to edit this.

Rephrasing of question: does there exist a non-constant holomorphic curve in any symplectic manifold? (presumably, for generic choice of compatible J) Can we say more in dimension 4? Are there conditions we can put on the symplectic manifold so that there exist curves?

• A generic K3 surface (which is a symplectic 4-manifold) does not have any curves, so the answer to the question in complete generality is "no". We therefore reinterpret the question to be about finding a large class of 4-manifolds for which we can say something.
• If we drop the non-constant condition, there are the trivial (constant) holomorphic curves. This is why we require non-constant holomorphic curves.
• If we allow ourselves to find a $J$-holomorphic curve for a very special (not generic!) almost complex structure $J$, it suffices to find an embedded symplectic surface and then construct $J$ to make this surface $J$-holomorphic. In dimension 4, we can find a symplectic surface by finding a Donaldson divisor.
• If there exists a $J$-holomorphic sphere in $N$, then there exist $J$-holomorphic maps from domains of all genus, by composing with a branched cover.
• There are two obvious infinite families of examples for which we can find non-constant holomorphic curves. The first are products of symplectic manifolds with surfaces. The second family of examples is obtained by blowing up a symplectic 4-manifold.
• Another family of examples come from 4-manifolds $(N, \omega)$ for which the Gromov-Taubes invariant is non-vanishing. For instance, if $c_1(TN) \ne 0$.
• If we allow ourselves to find a $J$-holomorphic curve for a very special (not generic!) almost complex structure $J$, it suffices to find an embedded symplectic surface and then construct $J$ to make this surface $J$-holomorphic. In dimension 4, we can find a symplectic surface by finding a Donaldson divisor.
• If there exists a $J$-holomorphic sphere in $N$, then there exist $J$-holomorphic maps from domains of all genus, by composing with a branched cover.
• Another family of examples come from 4-manifolds $(N, \omega)$ for which the Gromov-Taubes invariant is non-vanishing. For instance, if $c_1(TN) \ne 0$.