A speaker extraction algorithm seeks to extract the target speaker's speech from a multi-talker speech mixture. The prior studies focus mostly on speaker extraction from a highly overlapped multi-talker speech mixture. However, the target-interference speaker overlapping ratios could vary over a wide range from 0% to 100% in natural speech communication, furthermore, the target speaker could be absent in the speech mixture, the speech mixtures in such universal multi-talker scenarios are described as general speech mixtures. The speaker extraction algorithm requires an auxiliary reference, such as a video recording or a pre-recorded speech, to form top-down auditory attention on the target speaker. We advocate that a visual cue, i.e., lip movement, is more informative than an audio cue, i.e., pre-recorded speech, to serve as the auxiliary reference for speaker extraction in disentangling the target speaker from a general speech mixture. In this paper, we propose a universal speaker extraction network with a visual cue, that works for all multi-talker scenarios. In addition, we propose a scenario-aware differentiated loss function for network training, to balance the network performance over different target-interference speaker pairing scenarios. The experimental results show that our proposed method outperforms various competitive baselines for general speech mixtures in terms of signal fidelity.