The paper considers heat transport in diatomic one-dimensional lattices, containing equal amounts of particles with different masses. Ordering of the particles in the chain is governed by single correlation parameter -- the probability for two neighboring particles to have the same mass. As this parameter grows from zero to unity, the structure of the chain varies from regular staggering chain to completely random configuration, and then -- to very long clusters of particles with equal masses. Therefore, this correlation parameter allows a control of typical cluster size in the chain. In order to explore different regimes of the heat transport, two interatomic potentials are considered. The first one is an infinite potential wall, corresponding to instantaneous elastic collisions between the neighboring particles. In homogeneous chains such interaction leads to an anomalous heat transport. The other one is classical Lennard-Jones interatomic potential, which leads to a normal heat transport. The simulations demonstrate that the correlated disorder of the particle arrangement does not change the convergence properties of the heat conduction coefficient, but essentially modifies its value. For the collision potential, one observes essential growth of the coefficient for fixed chain length as the limit of large homogeneous clusters is approached. The thermal transport in these models remains superdiffusive. In the Lennard-Jones chain the effect of correlation appears to be not monotonous in the limit of low temperatures. This behavior stems from the competition between formation of long clusters mentioned above, and Anderson localization close to the staggering ordered state.
