Modified gravity appears in many guises. Originally restricted to the galactic scales in the form of the MOND paradigm, modified gravity in the dark energy community is understood as a modification of General Relativity on cosmological scale which could either modify the background evolution of the Universe or more appropriately the cosmological perturbations. In this case, both lensing and the growth of structure are sensitive to modified gravity effects and will be effectively sought for by the future generations of surveys.
Massive gravity also belongs to the modified gravity field of research. Massive gravity has had its ups in the last decades but also many downs such as instabilities of perturbation theory. The jury is still out on its status and prospects.