hello and welcome to health care magic,
I am Dr Sriram Nathan, an ENT surgeon and I shall try to help you out with your problems.
The lump of tissue which your daughter coughed up could be just a thick phlegm or a solidified mucous discharge. If she has not had any problems after this and her symptoms are recovering then you do not have to worry about.
If she is not getting better in a week or so then you must do these:\
1. Take a course of anti biotics like the amox-clav for five days in consultation with your doctor.
2. You may also add an antihistamine like levocetrizine
3. Take only warm water and avoid anything cold
4. Take warm water gargle with salt two to three times in a day
5. Avoid head bath for a couple of days till she recovers completely
If she does not get better in a week or so then you must take a throat swab and also chest X ray
and get back to me with the reports. You may show the specimen to the ENT surgeon although it is unlikely to be useful
Dr Sriram Nathan