Since the EM field is a linear combination of the electroweak $U(1)$ gauge field and one of the $SU(2)$ gauge fields, does this mean that it has self-interaction terms carried over from the electroweak $SU(2)$ kinetic term?
There is a term involving two factors of $A_\mu$ (the EM gauge field) and two factors of $W_\mu$ (the charged weak bosons). The Feynman diagram, as usually drawn, has four wavy lines meeting at a point; two of the lines are photons, and two are $W$-bosons. This Feynman diagram is shown in figure/equation (60) on page 12 in
- "A resource for signs and Feynman diagrams of the Standard Model," https://arxiv.org/abs/1209.6213.
For a source that writes out all the messy terms in the Lagrangian in terms of the $A_\mu$ gauge field and the $Z$ and $W$ gauge bosons, see equation (5.169) on page 249 in Renton's book Electroweak Interactions. The corresponding Feynman diagram is shown on page 574 in the same book.
However, there is no term involving more than two factors of the $A_\mu$ gauge field itself (no three-photon Feynman diagrams), which is what we usually mean by "self-interaction." So in that sense, the answer is no.