Factuality evaluation aims to detect factual errors produced by language models (LMs) and hence guide the development of more factual models. Towards this goal, we train a factuality evaluator, FenCE, that provides LM generators with claim-level factuality feedback. We conduct data augmentation on a combination of public judgment datasets to train FenCE to (1) generate textual critiques along with scores and (2) make claim-level judgment based on diverse source documents obtained by various tools. We then present a framework that leverages FenCE to improve the factuality of LM generators by constructing training data. Specifically, we generate a set of candidate responses, leverage FenCE to revise and score each response without introducing lesser-known facts, and train the generator by preferring highly scored revised responses. Experiments show that our data augmentation methods improve the evaluator's accuracy by 2.9% on LLM-AggreFact. With FenCE, we improve Llama2-7B-chat and Llama3-8B-chat's factuality rate by 16.86% and 14.45% on FActScore, outperforming state-of-the-art factuality finetuning methods by 8.83% and 6.96%.