The extension of DFPT to inlcude Hubbard U correction is described in the following papers:
A. Floris, S. de Gironcoli, E. K. U. Gross, M. Cococcioni, Phys. Rev. B 84, 161102(R), (2011);
A. Floris, I. Timrov, B. Himmetoglu, N. Marzari, S. de Gironcoli, and M. Cococcioni, Phys. Rev. B 101, 064305 (2020).
See example PHonon/example/example18/.