I have a lovely Murray who cork in the head joint won’t budge. Suggestions?
corkscrew
drop a dime on the end you want to push then jam it with the end of a wooden spoon
leave it where it is
Well, Leaving it where it is a great option of I want the 2nd oct to be outa tune (quite sharp) 
I’ll try the dime thing.
Anyone else???
more finesse would be okay, but the concept is dead on
a round metal thingy to keep the cork in one piece (hopefully)
a wooden dowel to push with that is long enough
drop coin in tendon end & follow with dowel
place other end on table and press the flute down
you may need a new cork
ya might want to oil the head joint while the cork is out
don’t use a hammer if you don’t have to 
I am no expert on this, but I would try a little bore oil at the cork/flute interface. I also would try tapping the cork with wooden dowel close to the diameter of the flute bore. The tapping may be able to unstick the cork without damaging it.
i’ve done all the above unsupervised and fortunately with success. but do at your own risk of course.
add alittle lube to the walls near cork
drop a dime over facecork
insert wooden dowel
tap dowel with mallet
dime/spoon worked! greased cork a wee bit. job done!
Thanks!!!