I am sorry to hear that your granddaughter has been unwell.
As her mother is asthmatic, she has a greater chance of having the same problem, at least as a child so you are right to point this out.
It is difficult to know if she is needing antibiotics without examining her as the green mucus you are reporting might be coming from the back of her throat.
However, as your granddaughter is wheezy, it is really best if she is taken to a doctor.
Do you have questions? Please feel free to ask.
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