есть cannot be used in a negative statement. For these types of simple statements, there are two formulas:
у + GEN + (есть) + NOM
Example: У меня есть машина.
Translation: I have a car.
Literally: At me (or near me) there is a car.
у + GEN + нет + GEN
Example: У меня нет машины.
Translation: I don't have a car.
Literally: At me (or near me) is not of car.
Another note on the inclusion/exclusion of есть. If you are introducing something that was previously unknown and the emphasis is on its EXISTENCE, use есть. If you are giving detail about something that is already known to exist, do not use есть. This is generally true for these types of statements, but not without exception.
A. У тебя есть машина? (does the car exist or not?) B: Да, у меня есть машина. (yes, it does exist) A: Какая машина у тебя? (what kind of car do you have?)
This is a grammatical example and people don't really speak with such complete ideas for such a simple conversation.
"Papa's not here" is grammatically equivalent to the translation you give and should not be considered wrong. Many English-speaking children say Papa (or Daddy) instead of Dad or Father. The apostrophe with an s at the end of a word is a conversational equivalent to the word is.