Minion running round holding key promises to be humiliating (11)
I believe the answer is:
'be humiliating' is the definition.
(I've seen this before)
'minion running round holding key promises' is the wordplay.
'running round' indicates anagramming the letters.
'holding' indicates putting letters inside.
'key' becomes 'g' (musical key).
'promises' becomes 'IOUs' (promises to pay a debt).
'minion' with letters rearranged gives 'inomin'.
'inomin' placed around 'g' is 'ignomin'.
'to' is the link.
(Other definitions for ignominious that I've seen before include "Deserving disgrace or shame" , "Abject" , "Causing disgrace" , "Deserving public disgrace" , "base" .)