Suppose a Lagrangian $L$ for a scalar field $\phi$, consists of a kinetic term and a Mexican-hat type potential. I am aware that if the vacuum has symmetry $H \subset G$ while the Lagrangian has symmetry $G \subset O(n)$, there will be symmetry breaking. By Goldstone's theorem there would appear dim $G$ - dim $H$ massless spinless bosons in a perturbation around the vacuum.
When the kinetic term dominates over the potential, meaning the norm of the derivatives of our field is larger than the value of the potential, does it still make sense to expand our field around the vacuum to extract information about the spectrum of particles of the theory? I guess the expansion has to be done on a stationary point of the Lagrangian with the same energy as our field and therefore not. In this case, how can one explain the Higgs mechanism?