  Motivated by examples that appeared in the context of string theory - gauge
theory duality, we consider corrections to supergravity backgrounds induced by
higher derivative R^4+... terms in superstring effective action. We argue that
supersymmetric solutions that solve BPS conditions at the leading
(supergravity) order continue to satisfy a 1-st order ``RG-type'' system of
equations with extra source terms encoding string (or M-theory) corrections. We
illustrate this explicitly on the examples of R^4 corrections to the
generalized resolved and deformed 6-d conifolds and to a class of non-compact
7-d spaces with G_2 holonomy. Both types of backgrounds get non-trivial
modifications which we study in detail, stressing analogies between the two
cases.
