Can food allergy and nutrition cause adhd symptoms.

No. Although you will find some doctors, who insist that they can cause ADHD symptoms, the standard, majority view -- and my view -- is that they can't.
Yes in many cases. As Dr. Smith noted, many Drs. are skeptical but it is clear that in many cases food sensitivities can cause ADHD symptoms. A recent study in Lancet found symptom improvement in 64% of children who avoided triggering foods. See Many nutrients like Fish Oil, B vitamins & magnesium often help too. Look for a B with the L-5-MTHF form of folate (folic acid).