What languages accept the use of mesoclisis and/or endoclisis?

Part of the problem is going to be that the terminology can get idiosyncratic to a language. I was not familiar with the terms endoclisis and mesoclisis, though I’m sure I’ve seen somewhere a description of an Italian dialect that sounds like what you’re describing as mesoclisis.

If we treat the Indo-European preverb as a separate word and not a prefix (which it seems to have been originally), some instances of mesoclisis show up in old Indo-European languages; Indo-European Language and Culture lists Old Irish, Gothic, and Avestan examples where a clitic comes between the preverb and the verb. In German now, just as in Homeric Greek, you can put a whole sentence between the preverb and the verb.

Endoclisis is an instance of Tmesis, where the interrupting word breaking up a word happens to be a clitic. Per Clitic – Wikipedia,

The endoclitic splits apart the root and is inserted between the two pieces. Endoclitics defy the Lexical Integrity Hypothesis (or Lexicalist Hypothesis) and so were long claimed to be impossible. However, evidence from the Udi language suggests that they exist. Endoclitics are also found in Pashto and are reported to exist in Degema.