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In this paper we will establish different weighted Poincar\'{e} inequalitieswith variable exponents on Carnot-Carath\'{e}odory spaces or Carnot groups. Wewill use different techniques to obtain these inequalities. For vector fieldssatisfying H\"{o}rmander's condition in variable non-isotropic Sobolev spaces,we consider a weight in the variable Muckenhoupt class $% A_{p(\cdot ),p^{\ast}(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses,and in this case we obtain the first order weighted Poincar\'{e} inequalitieswith variable exponents. In the case of Carnot groups we also set up the higherorder weighted Poincar\'{e} inequalities with variable exponents. For theseresults the crucial part is proving the boundedness of the fractional integraloperator on Lebesgue spaces with weighted and variable exponents on spaces ofhomogeneous type. Moreover, using other techniques, we extend some of theseresults when the exponent satisfies a jump condition and the weight is in asmaller Muckenhoupt class.  Finally, we will use these weighted Poincar\'{e} inequalities to establishthe existence and uniqueness of a minimizer to the Dirichlet energy integralfor a problem involving a degenerate $p(\cdot )$-Laplacian with zero boundaryvalues in Carnot groups.

Weighted $p(\cdot)$-Poincar\'e and Sobolev inequality]{Weighted $%  p(\cdot )$-Poincar\'{e} and Sobolev inequalities for vector fields  satisfiying H\"{o}rmander's condition and applications