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The shape described is a kite, which is a quadrilateral with two pairs of equal adjacent sides and one reflex angle. A kite has an axis of symmetry along the line connecting the midpoints of the non-congruent sides. The reflex angle in a kite is typically greater than 180 degrees, creating a distinct shape that is not a regular quadrilateral.
An arrowhead
A reflex angle
A four-sided shape with one right angle is a quadrilateral. There is no more specific term, unless it has more than one right angle or has equal sides.
There is no special name for such a shape.