Consistent use of personal pronouns

Question

Here is a excerpt from a dialog between a restaurant client and a waiter from an Italian source:

-- La minestra è fredda.
-- La prego di accettare le mie scuse. Gliene porto un’altra.

My question is: Are "La" in "La prego" and "Gliene" used correctly together in this discourse? To my understanding, "La prego..." implies the waiter is apologizing to a female client, and "Gliene porto..." implies that the waiter will bring the dish to a male client, which is contradictory. I think the dialogue will be correct if the waiter says: Lo prego di accettare le mie scuse. Gliene porto un’altra. Or perhaps my understanding of this matter is incorrect.

Answer 1

In Italian you can address someone informally (dare del tu) or formally (dare del lei).

Informal: Ti prego di accettare le mie scuse.
Formal: La prego di accettare le mie scuse.

"Ti" and "la" here are personal pronouns, and the sentences remain the same no matter the gender of the client.

For the same reason:

Informal: Te ne porto un’altra.
Formal: Gliene porto un’altra.

Again, the sentences remain the same no matter the gender of the client.

The following is not a complete explanation of formal Italian, but I wish it can help in this case:

Use of pronouns

The pronouns (direct and indirect, weak and strong) used in formal Italian are those of the third singular feminine person (lei, sé, la, le), even if the addressed person is a male. In this case, "la" and "gliene" ("le"+"ne") are used.

Verbs coniugation

If the person addressed is the subject, the coniugation of verbs is also the third singular person. In this case, the subject is always the speaker (the waiter).