We report a study of B→(J/ψγ)K and B→(ψ ′γ)K decay modes using 772×106 BB̄ events collected at the Υ(4S) resonance with the Belle detector at the KEKB energy-asymmetric e+e- collider. We observe X(3872)→J/ψγ and report the first evidence for χc2→J/ψγ in B→(Xcc̄γ)K decays, while in a search for X(3872)→ψ′γ no significant signal is found. We measure the branching fractions, B(B ±→X(3872)K±)B(X(3872)→J/ ψγ)=(1.78-0.44+0.48±0.12)×10-6, B(B ±→χc2K±)=(1.11-0.34+0. 36±0.09)×10-5, B(B±→X(3872)K ±)B(X(3872)→ψ′γ)<3. 45×10-6 (upper limit at 90% C.L.), and also provide upper limits for other searches.