Considering a large-scale Energy-Harvesting Wireless Sensor Network (EH-WSN) measuring compressible data, sparse random projections are feasible for data well-approximation and the sparsity of random projections impacts the Mean Square Error (MSE) as well as the system delay. In this paper, the authors propose an adaptive algorithm for sparse random projections in order to achieve a better tradeoff between the MSE and the system delay. With the energy-harvesting constraints, the sparsity is adapted to channel conditions via an optimal power allocation algorithm and the structure of the optimal power allocation solution is analyzed for some special case. The performance is illustrated by numerical simulations.