This paper advocates that power improvement based on the cross-fit variance estimator, proposed by Mikusheva and Sun (2022), is generally applicable to other econometric inference problems, where the statistics of interest are constructed by quadratic forms. We consider consistent specification testing for regression models, overidentifying restriction testing for linear instrumental variable regression models with many weak instruments, and parameter hypothesis testing with many weak instruments, develop the cross-fit variance estimators for those test statistics, and show that the resulting test statistics exhibit improved power properties. Numerical examples illustrate attractive finite sample properties of our cross-fitting approach.