Clayton is a known Traveller surname - but note that it is also a surname of the largest and best-known canal transportation company in England in the early C20 Fellows Morton & Clayton.
References to Romany Claytons have been found in DBY,HEF,LEI,STF and YKS between 1775-1965 - this from Bob Dawson's
A Romany in the Family published 2005.
See if your local library has a copy of
My Ancestors were Gypsies by Sharon Sillers Floate and have a read as that will give you lots of good background info.
Also have a look at the Romany and Traveller Family History Society website.
The Keith Chandler collection
https://rtfhs.org.uk/the-keith-chandler-collection-of-romany-and-traveller-biographies/ has a number of Clayton entries divided into three periods CLAYTON to 1830
CLAYTON to 1865 and CLAYTON to 1911
Look at the other things in the
https://rtfhs.org.uk/free-records/ section and you may find other useful information.
You might also want to ask for this thread to be moved to the Travelling People forum where there are lots of people with an interest in the Traveller community.
As for evidence what records do you have Frank, Abraham, William and John in? Look for mentions of traveller, sojourner, various Gypsy occupations, but of course at certain times of the year Gypsies could be agricultural labourers and living in houses not in tents or vardos. Look for an itinerant lifestyle with children being born or baptised over a wide area. Also look at the wives and their wider families too as these people often intermarried. If you find your family on a Census have a look at who is to be found nearby - they often travel in family groups so the surnames of others nearby can give clues to maiden names. They often use multiple names - depending on who is asking the question the reply may differ.
They are a fascinating group of people and you might also like to read
Travellers Through Time by Jeremy Harte or many of the other books available about them to learn more.
Hope this helps.
