Was looking at 1881 pub licensing and saw a Regent Place.
I can't see it on any maps I have.
I suspect the pub may be Plumbers Arms 32 Oxton Road ex book shop.
Other info
Birkenhead - Presbyterian church Regent Place, Grange Road West, Birkenhead, Cheshire.
This church was founded in 1848. It closed in 1976.
I think it may have been where the car park is at the back of the Little Theatre
Some of the "Places" in Birkenhead weren't roads they were rows of houses/buildings, examples nearby include Somerset Place and Vernon Place
Further to the above, the 1864 Morris Directory has quite a number of shops in Regent place with odd and even numbers going up to 23 (not all present).
Its looking like Clayton Place may have been Regent Place, as Clayton Place doesn't exist in that directory.
Could it have actually have been Grange Road West (which doesn't look like it's in the 1864 directory)
Regent Place was actually the name of the block of properties numbered 1-9 Grange Road West (then simply Grange Road, as modern Grange Road was then still Grange Lane) that stood between the Presbyterian Chapel and Charing Cross - see 1875 map below. I'm not sure why the name would occur in licensing records in 1881 though, as the Directory of that year lists the occupants of the properties as a tailor, a watchmaker, a baker/flour dealer, and a painter & plumber...
There's a few more than 9 addresses in Morris' Directory 1864
Bruce Mrs. John, 4, Regent place
Barnett John Alexander, wine and spirit merchant, 4 Regent place
Candler and Cole, grocers, tea dealers &c, 10 and 12 Regent place
Ellis Frederick, joiner and Cabinet maker, Regent place
Fletcher Thomas, baker and confectioner, 2 Regent place
Nisbet Miss Charlotte, milliner, 8 Regent place
Parsons Robert, tailor. 17 Regent place
Roberts Thomas H., bookkeeper, 23 Regent place
Stewart John, chemist and druggist, 6 Regent place
Sutton Jonathan, glass and china dealer. 19 Regent place
Wynn Saml., contractor, 15 Regent place
Possibly the other side of the road for the even numbers and the licence may have been for the Charing Cross Hotel.
I have been told that the Gold Shop ex Katech Computers used to be a chemist, but then again wasn't there a church on Charing Cross that was knocked down to build a bank