Would black and white cream with hydroquinone work on psoriasis area on my hand to relieve redness?

No. Black and white cream is an old, old, old patent remedy with little or no activity against psoriasis. Hydroquinone is used to lighten brown spots. There are many modern, effective treatments for psoriasis. See a dermatologist.
Discoloration. Other than the hydroquinone I am not familiar with the components of "black and white" cream. The hydroquinine can lighten pigment in some patients but the results can vary. As far as using it in psoriasis I am skeptical about how well it would work.