I just read this in a novel, and the sentence goes something like this
Perhaps she did have asthma
She did have. >>> You can only use the original form of the verb after the verb "do".
Grammatically, you can even say "she does do sth." >> Means "She indeed does sth."
While you may think
She +has asthma.
should be together, that is the present tense. However, your sentence is in the past tense
She had asthma.
When formed with did + infinitive root creates the *past
She did have asthma.
Auxiliary verbs in declarative sentences are used to emphasize the idea expressed by the verb, when the verb takes an auxiliary. Emphasis for tenses where auxiliaries are part of the sentence is heard through intonation only.
1) Perhaps she had asmthma. → Perhaps she did have asthma.
2) They read a lot of books last summer. → They did read a lot of books last summer.
3) They've been swimming a lot lately. → They have been swimming a lot lately.
In cases like this, the long form of the verb may be used in writing but when speaking, the auxiliary is stressed more if one wants to emphasize the idea about swimming.
4) We'd love to see them tomorrow. → We would love to see them tomorrow.
Again, the would cannot be shortened if you are emphasizing this with intonation. Otherwise, one would most likely shorten it. We'd etc. 5) They have a huge house. →They do have a huge house.