In this article, we study adaptation mechanisms in a class of phosphorylation cycles where allosteric binding and gene autoregulation mechanisms regulate the phosphorylation processes. We show that both mechanisms enable a robust setpoint regulation of the regulator metabolite in the presence of constant, as well as periodic, external stimuli. The allosteric binding mechanism without the presence of gene autoregulation can serve as an integral controller. Furthermore, we show that the incorporation of a gene autoregulation mechanism enables the gene expression system to act as a genetic oscillator, which allows for the adaptation mechanism to periodic external stimuli. These results provide a theoretical explanation to the cell homeostasis under quasi-constant environmental conditions, as well as periodic, biological rhythms.