We consider the \documentclass[12pt]{minimal}
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$$\end{document}=2 supersymmetric massive Yang-Mills field theory formulated in the \documentclass[12pt]{minimal}
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$$\end{document}=2 harmonic superspace. We present various gauge-invariant forms of writing the mass term in the action (in particular, using the Stueckelberg superfield), which result in dual formulations of the theory. We develop a gaugeinvariant and explicitly supersymmetric scheme of the loop expansion of the superfield effective action beyond the mass shell. In the framework of this scheme, we calculate gauge-invariant and explicitly \documentclass[12pt]{minimal}
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$$\end{document}=2 supersymmetric one-loop counterterms including new counterterms depending on the Stueckelberg superfield. We analyze the component structure of one of these counterterms.