SUMMARY
Please see the summary. I generally have the spellchecker on by default in the System Settings.
It seems a matter of font type: I mean, if I change it maybe I see no "error", but if I write some more words I have the all the new accented words underlined (even if I copy an "old" one). I don't have found a font type which is OK.

STEPS TO REPRODUCE
1. Use the Italian dictionary;
2. Open Kate (or KWrite) and write some words which end with an accented letter, for example "però, papà, così, dà";
3. They are all underlined (but they shouldn't :) ).

OBSERVED RESULT
In Italian, all the words which end with an accented letter are underlined (I can't tell you what would be if they were in the middle, in Italian they can be only at the end...).

EXPECTED RESULT
Only misspelled words should be underlined.

SOFTWARE/OS VERSIONS
Linux/KDE Plasma: openSUSE Tumbleweed 20210223
(available in About System)
KDE Plasma Version: 5.21.0
KDE Frameworks Version: 5.79
Qt Version: 5.15.2

ADDITIONAL INFORMATION
If I right click on a underlined word, for example "arriverà", and I choose "Spelling" from the popup menu I found "arriverà" in the suggested words, but if I click on it I'll have "arriveràà" (it seems to ignore the last letter).