We consider free scalar matter covariantly coupled to background W-gravity. Expanding to second order in the W-gravity fields, we study the appropriate anomalous Ward-Takahashi identities and find the counterterms which maintain diffeomorphism invariance and its W-analogue. We see that a redefinition of the vielbein transformation rule under W-diffeomorphism is required in order to cancel nonlocal contributions to the anomaly. Moreover, we explicitly write all gauge invariances at this order. Some consequences of these results for the chiral gauge quantization are discussed.