Bayesian nonparametric mixture models offer a rich framework for model-based clustering. We consider the situation where the kernel of the mixture is available only up to an intractable normalizing constant. In this case, the most commonly used Markov chain Monte Carlo (MCMC) methods are unsuitable. We propose an approximate Bayesian computational (ABC) strategy, whereby we approximate the posterior to avoid the intractability of the kernel. We derive an ABC-MCMC algorithm which combines (i) the use of the predictive distribution induced by the nonparametric prior as proposal and (ii) the use of the Wasserstein distance and its connection to optimal matching problems. To overcome the sensibility concerning the parameters of our algorithm, we further propose an adaptive strategy. We illustrate the use of the proposed algorithm with several simulation studies and an application on real data, where we cluster a population of networks, comparing its performance with standard MCMC algorithms and validating the adaptive strategy.