For Gamma = PSL2(Z) the hyperbolic circle problem aims to estimate the number of elements of the orbit Gamma z inside the hyperbolic disc centred at z with radius cosh(-1) (X/2). We show that, by averaging over Heegner points z of discriminant D, Selberg's error term estimate can be improved, if D is large enough. The proof uses bounds on spectral exponential sums, and results towards the sup-norm conjecture of eigenfunctions, and the Lindelof conjecture for twists of the L-functions attached to Maass cusp forms.