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The relationship between orbital and spin degrees of freedom in thesingle-crystals of the hole-doped Pr$_{1-x}$Ca$_{1+x}$MnO$_4$, 0.3 $\leq$ $x$$\leq$ 0.7, has been investigated by means of ac-magnetometry and chargetransport. Even though there is no cation ordering on the $A$-site, thequenched disorder is extremely weak in this system due to the very similarionic size of Pr$^{3+}$ and Ca$^{2+}$. A clear asymmetric response of thesystem to the under- (respective over-) hole doping was observed. Thelong-ranged charge-orbital order established for half doping ($x$=0.5) subsistsin the over-doping case ($x$ $>$ 0.5), albeit rearranged to accommodate theextra holes introduced in the structure. The charge-orbital order is howeverdestabilized by the presence of extra localized electrons (under-doping, $x$$<$ 0.5), leading to its disappearance below $x$=0.35. We show that in anintermediate under-doped region, with 0.35 $\leq$ $x$ $<$ 0.5, the``orbital-master spin-slave'' relationship commonly observed in half-dopedmanganites does not take place. The long-ranged charge-orbital order is notaccompanied by an antiferromagnetic transition at low temperatures, but by afrustrated short-ranged magnetic state bringing forth a spin-glass phase. Wediscuss in detail the nature and origin of this spin-glass state, which, as inthe half-doped manganites with large quenched disorder, is not related to themacroscopic phase separation observed in crystals with minor defects orimpurities.Comment: EPL style; 6 pages, 5 figure

Coexistence of long-ranged charge and orbital order and spin-glass state in single-layered manganites with weak quenched disorder