A child with alopecia, hyperpigmentation psoriatic dermatitis in genitals & mouth and hypogonadism is likely to be suffering from –
Features of Zinc deficiency include  :

Dwarfism (growth retardation) 
Diarrhea
Dermatitis
Hepatosplenomegaly
Iron deficiency anemia
Acrodermatitis enteropathica
Decreased immunocompetance
Poor wound healing
Hyperpigmentation
Hypogonadism
Perioral rash


Loss of hair and taste alterations may also be seen.

The patient in question presents with lzyperpigmentation, hypogonadisin, dermatitis and alopecia, all of which may be seen with Zinc deficiency.
Cu defeciency
Iron deficiency
Zn deficiency
Mg deficiency