The authors present results of a search for anomalous production of events containing a charged lepton (l, either e or [mu]) and a photon ([gamma]), both with high transverse momentum, accompanied by additional signatures, X, including missing transverse energy (E{sub T}) and additional leptons and photons. We use the same kinematic selection criteria as in a previous CDF search, but with a substantially larger data set, 305 pb−1, a p{bar p} collision energy of 1.96 TeV, and the upgraded CDF II detector. We find 42 l[gamma]E{sub T} events versus a standard model expectation of 37.3 ± 5.4 events. The level of excess observed in Run I, 16 events with an expectation of 7.6 ± 0.7 events (corresponding to a 2.7[sigma] effect), is not supported by the new data. In the signature of ll[gamma] + X we observe 31 events versus an expectation of 23.0 ± 2.7 events. In this sample we find no events with an extra photon or E{sub T} and so find no events like the one ee[gamma][gamma]E{sub T} event observed in Run I.