Have solved!
According to @obareey's comment, the key to the proof is trying to verify $P(LP)^\dagger$ is just the pseudoinverse of $LP$, using the existence and uniqueness of Moore-Penrose pseudoinverse. I am so pity that it's really a way I should find.
The following is the original question:
Let $ P\in \mathbb{C}^{n\times n}, L\in \mathbb{C}^{p\times n} $ and $P$ is a projection matrix, i.e. $P^2=P,P^H=P$, then why such equation holds $$ P(LP)^\dagger = (LP)^\dagger, $$ where $\dagger$ represents Moore-Penrose pseudoinverse.
The problem comes from a paper I read recently, whose name is A Weighted Pseudoinverse, Generalized Singular Values, And Constrained Least Squares Problem. It lies the end of the proof of Theorem2.1(Section2). The author say "it is easily seen that equation", while I couldn't understand the reason.
I have tried to focus on that P is a projection matrix and can be unitary similar diagonalized but I dosen't get the wanted result.