This would avoid mud on the Mendips, and give a choice of pubs and cafes.
Park at Shipham, head north and go straight on up the bridleway climb at the right hand bend in the road. Hit the tarmac, you could turn left and take the road to Tynings Farm. Past the race way and down the rocky road to Cheddar. Climb up through the gorge and up to the masts. Head west towards the trig point (this bit is a little muddy).
Climb to the trig point (which is muddy and rutted, but makes an interesting climb). Trig point down NE and then back towards the A38 direction on Limestone Link. It’s going to start to get muddy again after crossing the second stream. You could drop down towards Burrington then road back to Shipham, or use the fireroads in Rowberrow to get back to Shipham.
Not the most exciting route, but you will avoid the worst of the sludge.