Is there a bus or train service that runs from Manchester City to Mayfield in Staffordshire? My searches indicate it would involve an indirect route with changes.

If you haven't already, check using [http://www.transportdirect.info/web2/default.aspx?CurrentLanguage=English&&repeatingloop=Y], which I have always found to be very accurate.
If you don't have a look at any search engine and did not find any information then visit carlberry.co.uk were you may find some help.

Carlberry is sadly having trouble keeping up with bus changes - not always accurate these days
And yes - the journey does involve multiple changes and will take getting on for 3 hours
http://www.travelineeastmidlands.co.uk/em/XSLT_TRIP_REQUEST2?language=en&timeOffset=15
is another site to try - in the unlikely event of transport direct not coming up with the goods
