A search for a Higgs boson with suppressed couplings to fermions, h(f), assumed to be the neutral, lower-mass partner of the Higgs boson discovered at the Large Hadron Collider, is reported. Such a Higgs boson could exist in extensions of the standard model with two Higgs doublets, and could be produced via p (p) over bar -> H(+/-)h(f) -> W*h(f)h(f) -> 4 gamma + X where H (+/-) is a charged Higgs boson. This analysis uses all events withat least three photons in the final state from proton-antiproton collisions at a center-of-mass energy of 1.96 TeV collected by the Collider Detector at Fermilab, corresponding to an integrated luminosity of 9.2 fb(-1). No evidence of a signal is observed in the data. Values of Higgs-boson masses between 10 and 100 GeV/c(2) are excluded at 95% Bayesian credibility.