The Clayton copula is a mathematical tool used in copula theory to model dependence between random variables. It is a notable member of the Archimedean copula family and is best known for its ability to capture tail dependence. In this article, we present a new modified variant of the Clayton copula that aims to improve its flexibility. The proposed modification scheme perturbs its Archimedean nature by integrating a bivariate product of logarithmic functions and an additional tuning parameter. The elaborated copula benefits from a more nuanced representation of the copula density, and negative dependence can be obtained in a regular manner. We study its properties, including limit results showing some connection with the Gumbel-Barnett copula, important related functions, modifications and extensions, simulation of random couples of values, various lower and upper bounds, various tail dependences, and the correlation properties through the medial correlation and the Kendall tau. As an example of probability application, a new modified bivariate Gaussian distribution is presented via equations and graphics. Finally, two special cases of copula are discussed, including a simple single-parameter copula, which is intended to be a practical alternative to the Clayton copula. A brief analysis on simulated data shows that it may be preferable to the Clayton copula according to the Akaike information criterion. The overall result contributes to the advancement of the theoretical foundations of copula-based modeling techniques.