"What do you have for breakfast?"
Translation:Что у тебя на завтрак?
у тебя already means "you have", "у" there is a possessive preposition, it shows that a person possesses something. :) There есть isn't necessary.
Russian often leaves out the word "is" in the present tense, which can be confusing for English speakers because "есть" can mean "is" or "have" in different contexts. In this context, Russians would never say that you have something. They would say that something is for you and "есть" is optional, even unnecessary.
Because in Russian there is strict distinction between FOR+Recipient (для + Genitive case) and FOR+Circumstance (на + Accusative case). In your example, a breakfast is not a recipient of your gift or service, it is just a temporal circumstance.
According some charts of prepositions I've located, a noun which is the object of the preposition на is Accusative when на "means 'on to, on' when movement is involved" or Prepositional when it "means 'on, in, at' to define location".
на завтрак doesn't seem to fit either of these qualifications. There have been several exercises using на in Prepositional case when it was used to describe something going on on a bed, making the location of the activity the most significant part of determining case, so I'm not certain that "breakfast" could be interpreted as some kind of process - "breaking the fast" (what it literally means) by eating, or "breakfasting", engaging in eating breakfast, similar to the verbs "to dine" or even "to lunch".
This use of на seems extremely particular and limited in possible use - or is it more common than I am thinking?
In English, this use of "for" seems to involve a concept which includes activity. "For lunch" "for the holidays". Google Translate returned <<Мы идем в дом моих родителей на Рождество>> as a translation of "We go to my parents'house for Christmas"
In typing in the English text, I notice that Google was using для until I finished typing in "Christmas", at which point, Google changed для to на.
You are right, because using на + Accusative is not limited to the above; in this case, it is a circumstancial phraseologism. Another example: какие у тебя планы на эту неделю? - what plans do you have for this week?
Спасибо!!! Отлично!!! I think I understand this usage pretty well now. I only wish I could be as precise as you have been.
Surely Что ты ешь на завтрак is correct, as "have" means "eat" in this sentence, not "possess"?