This is obviously trickier than I first thought.Peterm and Joan would have normally raced through this by now.
To clarify the instructions..."the French" part is either la(x2) which was used at 1. and 6., le(x2), les(x2) and (used before a word beginning with a vowel) l(x1).
No.6 demonstrates how it should work, ie the two defns are Clean=launder/in accordance with=under - the wordplay indicates the longer of the defns LA-unde(nude dancing)-R and the answer is UNDER.
1.Ixion is feeble, first (l)ight (a)lways (m)ade (e)asy(2)(la)ME
2.Soft pedal for fear that badgers collective designation is protected by common market backing(4)
3.To go on irritating general perhaps swigged beer with tops off?(4)
4.They feel stress every second, half blind, yet they're relatively calm in a storm(4)
5.Dancer and company pull this expression of yearning to greatly amuse audience(4)
6.Clean in accordance with god, accepting nude dancing(5) (la)UNDER
7.See! Partners heavy blows boxing Queen, they rarely connect(5)
Gi(f)t w(r)ap (o)ld (g)uy(s), regularly insulting the French(5) FROGS