Can mupirocin heal or help with eczema?

Mupirocin is a topical antibiotic ointment used to treat bacterial skin infections.
Eczema is an inflammatory skin condition. Therefore mupirocin should not be used on eczema unless it has become bacterially infected, which would need to be diagnosed by a doctor.

Treatment of eczema will vary depending on the appearance (stage) of the eczema eg.acute "weeping" lesions, dry scaly lesions, or chronic dry, thickened lesions are each treated differently.

Talk with your pharmacist or doctor for more advice. See the link below.
https://www.drugs.com/eczema.html