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Under weak conditions assumed on the kernels, appropriate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula> estimates for certain class of generalized Marcinkiewicz integrals over subvarieties are established. By the virtue of the obtained estimates along with Yano’s extrapolation argument, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula> boundedness of the aforementioned integral operators under weaker conditions on the kernels is proved. The results in this paper represent essential extensions and improvements of many known results on the generalized Marcinkiewicz over symmetric spaces.