Anthony William Meddick - Help with finding my grandfather please

Hi all, I am looking for any help I can get with tracing my grandfather's roots. I only know his anglicised name and a rough DOB from the 1901C. I have a photo of him (looking about 20) with the stamp of Liverpool police constables office on it. I am presuming that must have been the port he arrived at, but don't know anything for a fact. He wasn't on the 1891C, but I have him & my grandmother on the 1901C living in the Broughton area of Manchester. I also have the marriage cert of his. He married in St Chads RC Church in Cheatham Hill, Manchester in 1899. His father is registered as Thomas & a farmer by proffession. My mother said he was from Poland, but the family said he was Lithuanian. Does anyone know which port he would have sailed from and am I correct in presuming Liverpool was a main port of arrival in those days?

I calculate that he arrived between 1892 & 1899. His anglacized name was Anthony WilliamMeddick. Many of the family said it was Medinski but I have checked that name against Polish/Lithuanian surnames & it doesn't match. Someone mentioned that it could have been Medeke, but as I said this is only guesswork. I would love to find out some information about him, but it looks pretty unlikely. Fingers crossed that at least I can get some input on the name and ports

thanks for any help given

Maybe one of these: Miedzik, Miedziński, Miedzicki?

Thank you for your suggestions, I think Miedzinski sounds very similar.

There are more than 1,000 people surnamed Medyński in Poland (the Medinski spelling is not Polish). The largest cluster is in SW Poland's Wrocław arrea, suggesting that the name's original ancetsral nest had been east of today's border. The port of departure for most Polish émirgrés in the olden hays were Bremerhafen and Hamburg. You've got very little to go on, so it's diffcult to glean more information from it.