[ABRIDGED]We study the carbon abundances with a twofold objective. On the one hand, we want to evaluate the behaviour of carbon in the context of Galactic chemical evolution. On the other hand, we focus on the possible dependence of carbon abundances on the presence of planets and on the impact of various factors (such as different oxygen lines) on the determination of C/O elemental ratios. We derived chemical abundances of carbon from two atomic lines for 757 FGK stars in the HARPS-GTO sample. The abundances were derived with the code MOOG using automatically measured EWs and a grid of Kurucz ATLAS9 atmospheres. Oxygen abundances, derived using different lines, were taken from previous papers in this series and updated with the new stellar parameters. We find that thick- and thin-disk stars are chemically disjunct for [C/Fe] across the full metallicity range that they have in common. Moreover, the population of high-$alpha$ metal-rich stars also presents higher and clearly separated [C/Fe] ratios than thin-disk stars up to [Fe/H],$sim$,0.2,dex. The [C/O] ratios present a general flat trend as a function of [O/H] but this trend becomes negative when considering stars of similar metallicity. We find tentative evidence that stars with low-mass planets at lower metallicities have higher [C/Fe] ratios than stars without planets at the same metallicity, in the same way as has previously been found for $alpha$ elements. Finally, the elemental C/O ratios for the vast majority of our stars are below 0.8 when using the oxygen line at 6158A however, the forbidden oxygen line at 6300A provides systematically higher C/O values. Moreover, by using different atmosphere models the C/O ratios can have a non negligible difference for cool stars. Therefore, C/O ratios should be scaled to a common solar reference in order to correctly evaluate its behaviour.
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