That's not IPA, at least not a standard form. So you should look at the source to figure out what they are trying to convey.

Of course /x/ is a voiceless velar fricative, so it might be indicating some of those properties. Velarization is usually expressed with a superscript of the voiced counterpart, as in: /tˠ/. But that usually only applies to consonants. I'm not quite sure what it would be for vowels (an extra-back pronunciation?).

Another possibility is that it is simply a "small" consonant, maybe not emphatic, maybe not released clearly, etc. Some non-standard transcriptions might do something like that, along the lines of indicating intonation information within a transcription.

Or is it possible you have the transcription slightly wrong? Could it be /e̽/? That's a mid-centralized vowel. Also an "x" but above the vowel, not as a separate superscript letter.

Older descriptions might have unusual conventions (not following IPA) or indicate non-systematic information (like a "small" consonant). If it's a new and otherwise-standard-IPA description, I'm not sure what else it would be adding. As I said, check the introduction or phonology section to see what they are trying to describe, if this is a grammar.

You could also look up the phoneme inventory and other phonological properties of the language on Wikipedia (or in another grammar) to see what it might be if you still can't figure it out.

Another option is just to search for the symbol itself online and see what comes up. Often Wikipedia has an entry for orthographic or transcription uses of symbols.

In this case, I think I found the answer that way:
https://en.wiktionary.org/wiki/Thread:User_talk:CodeCat/sign_%CB%A3_in_IPA
That links here: https://en.wiktionary.org/wiki/-e#Finnish

Just from the information there, it would appear that superscript /x/ represents an etymological blend of /-k/ and /-h/, which was my first interpretation. But that doesn't seem to be right. Instead, apparently superscript x represents gemination.

The article on Finnish phonology has:

Gemination or a tendency of a morpheme to cause gemination is sometimes indicated with a superscripted "x", e.g. vene /ʋeneˣ/. Examples of gemination:

https://en.wikipedia.org/wiki/Finnish_phonology#Sandhi

In my opinion, that's just a bizarre (mis)use of IPA. But I guess it's traditional for some research in Finnish etymology/philology? I have no expertise in that area so I can't comment further.

In short, it isn't a sound at all but a representation of the fact that a following consonant would be geminated (doubled/lengthened) when combined with this word. It seems that this actually is a historical remnant of the original /-k/ or /-h/ ending (see above), which was lost as an independent sound but preserved in contact with another consonant with which it assimilated.

This reminds me of Italian "redoppiamento sintattico" (or syntactic gemination/doubling) where an Italian word that in Latin had a final consonant may now trigger geminiation on a following word even though the consonant itself has been lost.
https://en.wikipedia.org/wiki/Syntactic_gemination
An example given there is "Andiamo a casa" ('we.go to home'), pronounced /...ak:asa/, with a geminated /k/, reflecting the fact that Italian 'a' comes from Latin 'ad', so that now-lost /d/ has an 'echo' just triggering gemination on the first consonant of a following word.

I guess something similar happens in Finnish and that is sometimes indicated with a superscript x!

(In defense of that approach, there isn't any other obvious way to do this in IPA so some additional notation is necessary. Another option would be something like /terʋe-:/, but that is confusing and doesn't quite convey what is intended. I've seen similar notation for tones applying to a following syllable/word though.)

Sounds exactly like redoppiamento sintattico, then, except that the etymology of the 'mystery consonant' may be unknown, while for Italian it is known (from Latin) but varies for different words.