Paul embrasse sa mère sur le cou avant de partir travailler.

In French, possessive adjectives (son, sa, ses) agree with the gender and number of the noun possessed, not the possessor.
Mère is feminine, so you must use sa mère, regardless of whether Paul is a man or a woman.
You would only use son with a masculine noun (e.g. son père).

In modern standard French, embrasser almost always means to kiss, not to hug.
For to hug, French speakers usually say prendre dans ses bras, faire un câlin, or simply câliner.
So here, Paul embrasse sa mère sur le cou is understood as Paul kisses his mother on the neck, not hugs.

French often uses the definite article (le, la, les) with body parts when the owner is already clear from the context.
Because we already know it is his mother, it’s obvious whose neck it is, so sur le cou is natural.
Sur son cou is possible but would put more emphasis on whose neck it is (e.g. to contrast with someone else’s neck), which is not needed here.

Sur le cou is the usual expression for a kiss on the neck.
Au cou can appear in some set phrases (e.g. tomber au cou de quelqu’un = to throw yourself into someone’s arms), but for a straightforward physical location like a kiss, sur le cou is the normal choice.

When avant is followed by a verb in the infinitive, French requires de: avant de + infinitive.
So you must say avant de partir, avant de manger, avant de sortir, etc.
Avant partir is incorrect in standard French.

There are two different structures:

• avant de + infinitive when the subject of both actions is the same (here: Paul kisses, Paul leaves).
• avant que + subject + subjunctive when the subject changes (e.g. avant que sa mère parte = before his mother leaves).
  Since Paul is both kissing and leaving, avant de partir is the correct form.

Grammatically, the subject of the infinitive partir is the same as the subject of the main verb embrasse, which is Paul.
So it means: Paul kisses his mother on the neck before he goes off to work.
If the mother were the one leaving, the sentence would need to be rephrased (e.g. avant que sa mère ne parte travailler).

In French, you can often chain two infinitives where English uses a preposition: partir travailler ≈ “leave to go work / leave for work.”
It’s a common pattern with verbs of movement, like venir voir, aller chercher, monter voir, etc.
Partir pour travailler is grammatical but doesn’t mean exactly the same thing; it sounds more like “to leave in order to work (somewhere),” stressing purpose rather than the everyday “go (off) to work.”

All three can be used in everyday French, but with slightly different flavors:

• partir travailler: “leave to go (off) to work,” quite natural and colloquial, focuses on the moment of setting off.
• partir au travail: “leave for work,” a bit closer to the English phrasing, and maybe a bit more concrete about the destination.
• aller travailler: “go to work” / “go work,” neutral and very common.

In many contexts they are interchangeable, though partir travailler emphasises the departure.

French often uses the present tense to talk about habits and repeated actions, where English frequently uses the simple present too, so here they match well: “Paul kisses his mother… before going to work.”
If you wanted to stress the habitual nature, you might add an adverb like tous les matins (every morning), but the present tense itself already allows a habitual reading.

The usual and most natural order is verb + direct object + other complements, so: Paul embrasse sa mère sur le cou.
Paul embrasse sur le cou sa mère is grammatically possible, but it sounds less neutral and slightly marked; it might appear in poetic or stylised language, or when you want to emphasize sur le cou.