It is no wonder you have been having difficulty finding it. There was never a pub in Longton called The Lamp. However, there was one that went under the name of The Red Lamp. It's proper name was The Duke of Wellington and it was located at 9 Bagnall Street (now Chelson Street). The building still exists. I presume your relative was the licensee of the premises in which case I might be able to help. I have records of all the Longton licensees going back to 1885. And then every ten years form the census returns back to 1841. It is mentioned in my book, The Lost Liquor Licenses of Longton that is still available.
Alan