Bonjour,

You can’t say “Peux-tu allumer ?”, wich only means “Can you turn on ?”.

“Peux-tu allumer la lumière ?” is perfect, it is the best translation for “turn on the light”.

It depends on the context. You can allumer all sorts of things. So you’d better mention what you are talking about.

Allumer une allumette.
Allumer une cigarette.
Allumer un feu.
Allumer un moteur/un appareil.
Not to mention all the figurative uses of the verb (allumer quelqu’un for example).

Imagine you come into a room which is dark and cold, the fireplace is nicely stacked with logs but the fire is not started. If you just say:

Peux-tu allumer ?

It might not be clear if you want the fire started or the light turned on, so you’ll say :

Peux-tu allumer la lumière ?

But if the room is getting dark and you say :

Il fait sombre, peux-tu allumer ?

it’s fine because mentioning Il fait sombre is enough to show you are talking about the light.

It is both correct and pleonastic.

If I enter in a dark room and I know that a friend is close to the light switch, I would probably ask him to switch on the lignt:

Tu peux allumer s’il te plait ?

(Tu peux is here a familiar interogative form, the correct one would be Peux tu)

But if I was asking some one with more respect, I would probably say:

Bonjour, pouvez vous allumer la lumière s’il vous plait

simply to be more explicit