In many biological, psychological or medical studies, two samples are observed. The aim of these studies is making inferences among the treatment effects involved in the trial. Hereby, other variables that are called covariates can obscure the factor effects. Analysis of Covariance (ANCOVA) can adjust the treatment effects for the impact of the covariates on the response variable. The classical ANCOVA model is regularly based on the assumptions of multivariate normality and equal variances. However, these assumptions are frequently hard to be justified. Inference with violation of the assumptions may lead to conservative or liberal test decisions. We modify the well-known Welch's t-test to covariate adjustments, which neither assume homogeneous variances across the treatment groups nor normal distributions. Simulation studies show that the new test with the t-approximation controls the type-1 error rate for small sample sizes and/or unbalanced designs. Comparison between the new method and other related solutions, like the involvement of HC0-HC3 estimators, demonstrates that our new method is competitive. The application of the proposed methods is illustrated by a real data set.