Does there exist a word with H sounds like [eɪtʃ]?

Solution 1:

The answer would have to be no in today's English. However the noun ache used to be pronounced /eitʃ/, like the letter H, contrary to the verb to ache that had a /k/ consonantal sound. The OED remarks on it:

Old English æce is a primary derivation of the verb ac-an to ache /eik/, in which, as in parallel forms, the c /k/ was palatalized to ch /tʃ/, while in the verb it remained /k/; cf. make — match; bake — batch; wake — watch; break — breach; speak — speech; stick — stitch.

Occasional early instances of ake as noun are northern, in which dialect c /k/ was not palatalized, cf. make = match, steik = stitch, kirk = church.

In the 17th century the noun was still atche /ɑːtʃ, ɛːtʃ/, plural atch-es /ɑːtʃɪz, ɛːtʃɪz/, but about 1700 it began to be confused with the verb as /ɛːk/. The spelling of the latter has in turn been changed to ache, so that though both verb and noun are now really ake, both are in current spelling written ache. The former pronunciation survives in the dialectal eddage = head-ache.

The ‘O.P.’ rioters, ignorant of the Shaksperian distinction of ache /eik/ (verb) and ache /eitʃ/ (noun), ridiculed the stage pronunciation of the noun by giving it to the verb in ‘John Kemble's head aitches’.

For the O. P. riots, see https://en.wikipedia.org/wiki/Old_Price_Riots

Shakespeare plays on the fact that H and ache were pronounced the same in Much Ado about Nothing (Act III, Scene IV):

Beatrice: 'Tis almost five o'clock, cousin, 'tis time you were ready. By my troth, I am exceedingly ill, heigh-ho!

Margaret: For a hawk, a horse, or a husband?

Beatrice: For the letter that begins them all, H.

A further example is taken from John Heywood's Hundred Epigrammes (1550), the 59th of which is about the letter H and works only if there is an identity of sound with the noun ache:

'H' is worst among letters in the cross row,

For if thou find him other in thine elbow,

In thine arme, or leg — in any degree —

In thy head, or teeth, in thy toe or knee,

Into what place soever 'H' may pike him,

Wherever thou find ache, thou shalt not like him.

Mention should also be made of the aitch-bone /ˈeɪtʃˌbəʊn/, defined in the OED as follows:

The bone of the buttock or rump; the cut of beef lying over this bone.