Yes, mortal men and women can worship Poseidon.
To be a god.
Amymone's son by Poseidon was called Nauplius.
His mortal sons were Theseus, Pelias, Neleus and Hippothoon.
A demigod is half mortal and half god, meaning one parent is god and one parent is mortal. a demigod of Poseidon is a demigod with Poseidon as his/her father and a mortal mother. an example of this is Percy Jackson in the "Percy Jackson and the Olympians Series" By Rick Riordan
there is not record of Poseidon ever having any mortal daughters (half bloods ) at all ,his female offspring were either imortal,nympes,giants and even animal's mortal children were men....
When they wished.
He was the son of Poseidon and a mortal woman.
Triton, Athena, Despoine, Proteus, Aiolos, and Asopos are Poseidon's children. Poseidon also has nymphe, animal, giant and mortal offspring.
As many immortal/mortal/demigod daughters as Poseidon has.
They are forbidden to have children with mortal women.
34: see link below.
Poseidon was an ancient Greek god. Some modern Pagans also worship him.