Using grand canonical Monte Carlo simulations of a polyelectrolyte chain where the charges are in contact with a reservoir of constant chemical potential given by the solution pH, we study the behavior of weak polyelectrolytes in a poor solvent. We address the influence of variable screening of the electrostatic interaction on the chain structure for rather poor solvents as well as in the close-to-Theta-point regime. For the latter case, we demonstrate that a fine tuning of the chain structure is attainable by varying screening. With growing screening length, that is, by reducing the salt concentration, the degree of charging is reduced and the pearl distribution can be shifted toward a smaller number of pearls. In addition, we find that pearl necklaces can also occur at high charging provided the screening is very strong. In the rather-poor-solvent regime, the position of the discrete transition from a highly charged, stretched chain to a weakly charged globule can be shifted by the variation of screening. By reducing the screening of electrostatic repulsion, we observe in both regimes a nonmonotonic stretching of the polyelectrolyte. After reaching a maximum of stretching, the chains relax back to an asymptotic value at vanishing screening, which depends on pH. For rather poor solvents, at large screening lengths, the chains collapse into an almost uncharged globule provided that high ionization is not pinned by a large chemical potential of the charges.