Random projection can reduce the dimension of data while capturing its structure and is a fundamental tool for machine learning, signal processing, and information retrieval, which deal with a large amount of data today. RandNLA (Randomized Numerical Linear Algebra) leverages random projection to reduce the computational complexity of low-rank decomposition of tensors and solve least-square problems. While the computation of the random projection is a simple matrix multiplication, its asymptotic computational complexity is typically larger than other operations in a RandNLA algorithm. Therefore, various studies propose methods for reducing its computational complexity. We propose a fast mixed-precision random projection method on NVIDIA GPUs using Tensor Cores for single-precision tensors. We exploit the fact that the random matrix requires less precision, and develop a highly optimized matrix multiplication between FP32 and FP16 matrices -- SHGEMM (Single and Half-precision GEMM) -- on Tensor Cores, where the random matrix is stored in FP16. Our method can compute Randomized SVD 1.28 times faster and Random projection high order SVD 1.75 times faster than baseline single-precision implementations while maintaining accuracy.