Automatic radiology report generation is challenging as medical images or reports are usually similar to each other due to the common content of anatomy. This makes a model hard to capture the uniqueness of individual images and is prone to producing undesired generic or mismatched reports. This situation calls for learning more discriminative features that could capture even fine-grained mismatches between images and reports. To achieve this, this paper proposes a novel framework to learn discriminative image and report features by distinguishing them from their closest peers, i.e., hard negatives. Especially, to attain more discriminative features, we gradually raise the difficulty of such a learning task by creating increasingly hard negative reports for each image in the feature space during training, respectively. By treating the increasingly hard negatives as auxiliary variables, we formulate this process as a min-max alternating optimisation problem. At each iteration, conditioned on a given set of hard negative reports, image and report features are learned as usual by minimising the loss functions related to report generation. After that, a new set of harder negative reports will be created by maximising a loss reflecting image-report alignment. By solving this optimisation, we attain a model that can generate more specific and accurate reports. It is noteworthy that our framework enhances discriminative feature learning without introducing extra network weights. Also, in contrast to the existing way of generating hard negatives, our framework extends beyond the granularity of the dataset by generating harder samples out of the training set. Experimental study on benchmark datasets verifies the efficacy of our framework and shows that it can serve as a plug-in to readily improve existing medical report generation models. The code is publicly available at https://github.com/Bhanu068/ITHN.