Here is the transcription from the book in the Bear Family box set 'Darby and Tarlton: Complete Recordings'.

I'm Irish police, preserving the peace
I belong to the Broadway patrol
Don't I look grand with a club in my hand
Every day out for a stroll

A darling in love from the window above
'Kind Charlie, won't you come here?'
'Oh Dolly, you're clover, how happy all over
I belong to Patrick O'Flynn

If I catch that bum a-loaded with rum
I'll cripple and sure run him in
I'm Irish police, preserving the peace
I belong to the Broadway patrol

Each darling I meet a-walking these streets
I arrest 'em and ride 'em with me
I'm the Irish police, preserving the peace
I belong to the roadway patrol

Parts of that transcription make no sense. 'How happy all over' is ridiculous. I agree with cnd and Modette that it should be 'I'll arrest you all over'. Like them, I hear 'Patrick-o clan'. I reckon it is simply saying that he is an Irish catholic who frowns upon prostitution. In the third stanza, it should be 'a bum' not 'that bum', suggesting that dealing with public drunkenness is another of his patrol activities.

--Stewie.