Extend OTP on random data?
No, that doesn't work. OTP is secure because knowledge of the plaintext doesn't give you any useful information about the key. This is because the bits of the key (e.g. $P_0$) are never used to encrypt anything else.

If you would somehow reuse the key then leakage of the plaintext would cause leakage of $P_0$. Leakage of $P_0$ directly leaks $P_1$. I.e. instead of perfect security the information encrypted by $P_1$ is now directly dependent on the security of the plaintext and $P_0$.

OTP is only perfectly secure as long as the bits in the key stream are independent of each other. OTP doesn't allow for shortcuts in that respect.