Is there any advantage using this kind of ROM-based randomized hashing in digital signatures? The said randomized hasing is actually a sort of padding, and it goes like this:
Parameters: 
1. hash function H()
2. length N in bytes of the output of the hash function.
Input: 
1. M the message to hash.
Generate:
1. Generate r := N/2 random bytes.
2. Compute lo := H(r|M)
3. Compute hi := r xor lo
4. _Return (hi|lo)
Verify:
1. Compute r := hi xor lo
2. Compute v := H(r|M)
3. _Return v == lo

Is there any advantage to it other than slightly better verification of the correctness of the padding?
You are essentially describing a salted hash which is very important for some post-quantum signature schemes. In some lattice-based schemes it is important that the same signing key never be used with the same hash so that some message-replay defence is required. Schemes such as FALCON achieve the anti-replay by salting the message before hashing. Your method is one way in which this could be achieved.