Automatic speech recognition systems deteriorate in presence of overlapped speech. A popular approach to alleviate this is target speech extraction. The extraction system is usually trained with a loss function measuring the discrepancy between the estimated and the reference target speech. This often leads to distortions to the target signal which is detrimental to the recognition accuracy. Additionally, it is necessary to have the strong supervision provided by parallel data consisting of speech mixtures and single-speaker signals. We propose an auxiliary loss function for retraining the target speech extraction. It is composed of two parts: first, a speaker identity loss, forcing the estimated speech to have correct speaker characteristics, and second, a mixture consistency loss, making the extracted sources sum back to the original mixture. The only supervision required for the proposed loss is speaker characteristics obtained from several segments spoken by the target speaker. Such weak supervision makes the loss suitable for adapting the system directly on real recordings. We show that the proposed loss yields signals more suitable for speech recognition and further, we can gain additional improvements by adaptation to target data. Overall, we can reduce the word error rate on LibriCSS dataset from 27.4% to 24.0%.