In this paper, we consider an impulsive mixture noise process, which commonly comes across in applications such as multiuser radar communications, astrophysical imaging in the microwave range and kick detection in oil drilling. The mixture process is in the time domain, whose probability density function (PDF) corresponds to the convolution of the components' PDFs. In this work, we concentrate on the additive mixture of Gaussian and Cauchy PDFs, the convolution of which leads to a Voigt profile. Due to the complicated nature of the PDF, classical methods such as maximum likelihood estimation may be analytically not tractable; therefore, to estimate signals under such noise, we propose using a Markov chain Monte Carlo method, in particular the Metropolis-Hastings algorithm. For illustration, we study the estimation of a ramp function embedded in the Cauchy-Gauss mixture noise, which is motivated by the kick detection problem in oil drilling. Simulation results demonstrate that the mean square error performance of the proposed algorithm can attain the Cramer-Rao lower bound.