Xjyuk

There seem to be certain words in Latin which start with an underlying /gn/, such as noscō /gnosko:/ [nɔsko:]—this "hidden" /g/ appears when prefixes are added, as in cognoscō /congnosko:/ [cɔŋnɔsko:] (with the prefix con-).

If we see a word that starts with ign-, is that a sure sign of such an "invisible" /g/? For example, can we say with confidence that ignāvus, ignōbilis, and ignōrō have an underlying /g/ in their stems?

EDIT: I'm particularly interested in words that show clear prefixing—ignis, for example, doesn't count.

The short answer would be no. Nomen is a well-known example of a word that did not historically start with a velar consonant but that has a velar in some related prefixed words: agnomen, cognomen, ignominia (but not in praenomen or pronomen). This is thought to be the result of analogy.

There seem to be certain words in Latin which start with an underlying /gn/

I wouldn't put it this way. Words like nosco* historically started with a velar consonant. It's much less obvious that an "underlying" /g/ continued to be present at the start of words like nosco after they came to be pronounced with initial [n], and spelled with initial <N>.

This topic is covered extensively in Aspects of the Phonology and
Morphology of Classical Latin, by András Cser (2016). See Chapter 11, "The issue of〈gn〉-initial stems" (pp. 194-205).

Cser suggests a phonological development from a form with gn, to a form with a dorsal "floating C-Place node" before n, to a form with initial n:

it is clear that the literary period saw the gradual disappearance of the floating C-Place node and the lexical split of words in which it occurred. The unprefixed forms were relexicalised with a single initial [n], whereas the prefixed forms were relexicalised with a fully specified [ŋn] sequence which was no different from the [ŋn] sequence found internally in the regnum and
ignis-type words, and from this point on the relation between these unprefixed
and prefixed forms was no longer motivated phonologically. This made it possible
for other prefixes to attach to 〈gn〉-initial (now phonologically [n]-initial) stems, hence the novel formations like renatus ’born again’, praenoscere ‘know in advance’, pernobilis ‘most noble’.

(p. 202)

*I think the o in the first syllable is long /o:/—thought to come in this word from a vowel + *h₃ sequence, according to de Vaan—although I don't know what direct evidence there is of the length.