How do you know the words must be misspelt, even if you’ve never come across them before? Answer: because they’re of Greek etymology, and Greek double rr is always followed by h. Think of catarrh, myrrh and diarrh(o)ea.
So, of course, is Greek word-initial r: we write rhapsody, rheostat, rhesus, rhetoric, rheumatism, rhinoceros, rhizome, rhododendron, rhombus, rhubarb and rhythm, all being words derived from Greek; and there are no Greek-derived words in English that begin with plain r.
In Wikipedia we read that a mycorrhiza is a symbiotic association between a fungus and the roots of a vascular plant.
Why do Greek words have these apparently superfluous h? It can be traced back to an allophonic fact about Ancient Greek /r/, namely that it was voiceless in word-initial position and when geminated.
Generally speaking [r] is a voiced sound, but in certain environments in classical Attic it seems to have been voiceless. What we are actually told by the grammarians is that ρ was ‘aspirated’ at the beginning of a word, and that when a double ρρ occurred in the middle of a word the first element was unaspirated and the second aspirated. … These descriptions are followed in the Byzantine practice of writing initial ῥ and medial ῤῥ, and are supported at an earlier period by Latin transcriptions such as rhetor, Pyrrhus… [W.S. Allen, Vox Graeca, CUP 1968]
Allen goes on to say that it can be shown to have been a matter of voiceless r̥ (rather than a cluster rh) by the fact that it was borrowed into Latin on the one hand as rh but into Armenian on the other hand as hռ hṙ, thus hռետոր hṙetor, and similarly in Coptic.
When I did Greek at school, the transition from the fourth form into the fifth was marked by requiring us thenceforth to write the proper polytonic accents on our Greek, which we had until then been allowed to ignore. So every initial rho had to be written with a rough breathing, and every doubled rho with first a smooth breathing and then a rough.
The Greeks themselves did not bother with breathings and accent marks until the Hellenistic period. They first appear in papyruses in the second century AD, and then only sporadically.
It seems a bit silly that now, well over a thousand years later, we have to go on including this unnecessary letter h after r in English spelling: but that’s the power of tradition.
The Greeks themselves officially abandoned polytonic orthography, including the breathing marks, in 1982. The rough breathing, in the shape of the letter h, lives on only in English, French and German.
If the second part of a compound word in Greek began with r, the consonant was automatically doubled. That’s why we have one r in rheostat but two in diarrhoea. And one in rhizome but two in mycorrhiza.