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This paper studies a new class of instantaneous and non-instantaneous impulsive boundary value problem involving the generalized <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Caputo fractional derivative with a weight. Depending on critical point theorems and some properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-Caputo-type fractional integration and differentiation, the variational construction and multiplicity result of solutions are established.