You’ll find the remains by going to the end of Old Quay Lane and following the foot-path across the field, over the stream right down to the marsh.
Just a few bricks on an overgrown site down by the marshes remain now from the house that was built sometime around the 1620s or 30s. It was an inn in the late 1660s, the landlord was also a mariner and involved in shipping cattle from Dublin.
Some of the important guests in its heyday as an inn in the 1680s included the Lord Lieutenant of Ireland and the Earl of Derby.
In 1711 the building was divided into three dwellings; it was no longer in demand as an inn, as the Quay at that end of the Neston anchorage fell into disuse.
In 1750 the owner leased it to Cheshire County and they built a lavatory block and used it as the HOUSE OF CORRECTION where the destitute seasonal Irish harvest workers were housed awaiting a sailing to Dublin. Over 25,000 were shipped back over the next fifty years.
The Quay House became a private residence in the early 1800s and the artist Henry Melling made his home there until about 1875; he had 250 of his paintings displayed around the house at one time. The next tenant ran it as a refreshment room. It was then a farm, and the building was almost totally demolished in World War II.
Lots of detail in this article from 1930: https://www.hslc.org.uk/…/uploads/2017/06/79-9-Rideout.pdf