The closest to this idiomatic phrase I could find was:
quaeso, quid narras?
sorry, what are you saying?
Terence, Phormio, act 5, scene 8
Which, in this particular dialogue, is not quite “sorry, I didn’t hear that, could you repeat it?” but more a rhetorical “excuse me, what on earth are you saying?” However, it does seem that quaeso + verb of speaking/repeating/telling certainly does encapsulate that casual, almost throwaway “excuse me/pardon/what was that?” that we use in English.
Another similar example is the following:
dic dum, quaeso, mi, es tu Myconius?
excuse me, tell me, are you from Myconos?
Terence, The Mother-in-Law, act 5, scene 3
This is very different from ignosce mihi which is often used more like “pardon me, I’ve done something I really hope you will overlook”. Here it is used in a casual way like the quaeso above, yet the tone/meaning is slightly different:
tunc tunc—ignoscite (nolo, quid faciam?) sed sum petulanti splene—cachinno
then, then - excuse me (I don’t want to, what can I do?) but I have a cheeky temper – I cackle
Persius, Satires, 1
quisquis es, ignoscas; in nullam lumina partem
gurgite ab hoc flexi studioque operatus inhaesi
whoever you are, please forgive me; I haven’t taken my eyes off this pool so intent am I on fishing [and therefore can't help you in your search]
Ovid, Metamorphoses, 8.864–5
Since posting my original answer, I came across the following two examples that you might find useful for your context (speaking a second language).
Phormia: cena “dubia” apponitur
the “problematic” meal is set before you
Geta: quid istuc verbist?
what sort of expression is that?
Terence, Phormio, act 2, scene 2
Hanno: whispers something in an aside
Milphio: quid ais tu?
what are you saying?
Plautus, Poenulus, act 5, scene 2
(This is actually a very funny scene where Hanno, a Carthaginian, speaks Punic and Agorastocles pretends to translate for his master, Milphio)
I don’t think either phrases are particularly polite, but perhaps if you just added a quaeso, it would soften it?