sick headache vs head ache

sick headache and head ache both are nouns.

Word	Noun	Adjective	Verb	Adverb
sick headache	Yes	No	No	No
head ache	Yes	No	No	No

As nouns, head ache is a hypernym of sick headache; that is, head ache is a word with a broader meaning than sick headache:

sick headache: a severe recurring vascular headache; occurs more frequently in women than men
head ache: pain in the head caused by dilation of cerebral arteries or muscle contractions or a reaction to drugs

Other hypernyms of sick headache include cephalalgia, headache.

As nouns, head ache is a hypernym of sick headache; that is, head ache is a word with a broader meaning than sick headache:

sick headache: a headache accompanied by nausea
head ache: pain in the head caused by dilation of cerebral arteries or muscle contractions or a reaction to drugs

Other hypernyms of sick headache include cephalalgia, headache.

sick headache (noun)	head ache (noun)
a severe recurring vascular headache; occurs more frequently in women than men	pain in the head caused by dilation of cerebral arteries or muscle contractions or a reaction to drugs
a headache accompanied by nausea