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In this paper, we consider image quality assessment (IQA) as a measure of how images are amenable with respect to a given downstream task, or task amenability. When the task is performed using machine learning algorithms, such as a neural-network-based task predictor for image classification or segmentation, the performance of the task predictor provides an objective estimate of task amenability. In this work, we use an IQA controller to predict the task amenability which, itself being parameterised by neural networks, can be trained simultaneously with the task predictor. We further develop a meta-reinforcement learning framework to improve the adaptability for both IQA controllers and task predictors, such that they can be fine-tuned efficiently on new datasets or meta-tasks. We demonstrate the efficacy of the proposed task-specific, adaptable IQA approach, using two clinical applications for ultrasound-guided prostate intervention and pneumonia detection on X-ray images.