[Part I, cf. ibid. 112, 1-8 (1993; Zbl 0765.11038).] This is the second in a series of papers developing the authors' method for bounding automorphic \(L\)-functions at a point on the critical line. Such \(L\)-functions possess a functional equation due to Hecke which may be combined with the Phragmen-Lindelöf principle to yield the so-called convexity bound and the problem under consideration is to improve on this. Such an improvement will often have interesting consequences. There are a number of parameters in which these \(L\)-functions vary. In this paper the authors study, at a fixed point, the \(L\)-functions \(L_ f\) of (normalized) holomorphic cusp forms (of fixed weight \(k\geq 2\)) of the congruence group \(\Gamma_ 0 (q)\) with respect to the level \(q\). In this case the convexity bound is \(\ll q^{1/4+ \varepsilon}\) and the authors succeed in replacing \({1\over 4}\) by \({1\over 4}- \delta\) with a small \(\delta\) (\(\delta= {1\over {192}}\) always, and in some cases a little better). The case \(k=2\) includes the \(L\)-functions of modular elliptic curves. The method requires one to deduce the bound in question in trivial fashion from that for a certain mean square average of the type \(\sum\limits_ f | L_ f|^ 2\) where the sum is over a basis for the space of cusp forms one of whose members is the form in question. The paper contains a number of such mean value theorems which may be of independent interest. A necessary auxiliary result, which is a natural generalization of the classical `additive divisor problem' appears in [ibid. 115, No. 2, 209-217 (1994; Zbl 0791.11049)]. At the end of the paper, a short note corrects an erroneous step in the first paper of the series.