  We consider higher-derivative perturbations of quantum gravity and quantum
field theories in curved space and investigate tools to calculate counterterms
and short-distance expansions of Feynman diagrams. In the case of single
higher-derivative insertions we derive a closed formula that relates the
perturbed one-loop counterterms to the unperturbed Schwinger-DeWitt
coefficients. In the more general case, we classify the contributions to the
short-distance expansion and outline a number of simplification methods.
Certain difficulties of the common differential technique in the presence of
higher-derivative perturbations are avoided by a systematic use of the
Campbell-Baker-Hausdorff formula, which in some cases reduces the computational
effort considerably.
