Machine unlearning has garnered significant attention due to its ability to selectively erase knowledge obtained from specific training data samples in an already trained machine learning model. This capability enables data holders to adhere strictly to data protection regulations. However, existing unlearning techniques face practical constraints, often causing performance degradation, demanding brief fine-tuning post unlearning, and requiring significant storage. In response, this paper introduces a novel class of machine unlearning algorithms. First method is partial amnesiac unlearning, integration of layer-wise pruning with amnesiac unlearning. In this method, updates made to the model during training are pruned and stored, subsequently used to forget specific data from trained model. The second method assimilates layer-wise partial-updates into label-flipping and optimization-based unlearning to mitigate the adverse effects of data deletion on model efficacy. Through a detailed experimental evaluation, we showcase the effectiveness of proposed unlearning methods. Experimental results highlight that the partial amnesiac unlearning not only preserves model efficacy but also eliminates the necessity for brief post fine-tuning, unlike conventional amnesiac unlearning. Moreover, employing layer-wise partial updates in label-flipping and optimization-based unlearning techniques demonstrates superiority in preserving model efficacy compared to their naive counterparts.