We propose a novel scheme based on compressive sensing and sparse recovery to boost the performance of cross-packet random linear coding (RLC) by incorporating the partial packets in the decoding algorithm. In conventional RLC schemes, to successfully decode the packets the receiver needs to collect a certain number of correct innovative encoded packets. During this process, there are usually a lot of partially correct packets that are discarded. Our objective is to recover the errors in the partial packets to decrease the total transmitted packets to improve the performance in terms of throughput and energy efficiency. Assuming a systematic RLC, we first formulate this problem in form of a standard sparse recovery problem where the channel errors are sparsely distributed within the packets. Then we show that to correct a certain number of errors at the receiver, the minimum required number of transmitted packets is lower-bounded by the number of partial packets. We show that by correcting and exploiting the partial packets, the required number of RLC transmit packets to successfully deliver a given generation is reduced by typically 57% in comparison with the conventional scheme.