It might be worth you putting a post on Sheffield Forum in the History & Expats section to help you identify the Tea Garden. You'll get a wider audience for this type of search (lots of similar searches on there).
Hello, you may be remembering Marsden's Tea Rooms in Apperknowle. People visited from Chesterfield and Sheffield as a day out. It may have closed around the late 1960's. I believe the café was run by two retired nurses, it was a very smart affair and had a conservatory. Sorry I have not more information but maybe Dronfield library could help you. J Wilson
The tea room was in Apperknowle .I used to call in with my two children when walking from New Whittington round the block as we used to say.I would push them all the way in a push chair from New Whittington through the woods to the top road from Apperknowle to Old Whittington and back to New Whittington.The tea room was near where the TV mast now stands.I must have been very fit in those days pushing two babies(12mths apart in age)in one pushchair all that way and through the woods.This was the late 60's.I remember they had two or three squirrel drays in the trees in the garden as the children were always looking out for them.Happy Days.!!!