  The gas-solid budget of carbon in protoplanetary disks is related to the composition of the cores and atmospheres of the planets forming in them. The key gas-phase carbon carriers CO, C$^{0}$ and C$^{+}$ can now be observed in disks. The gas-phase carbon abundance in disks has not yet been well characterized, we aim to obtain new constraints on the [C]/[H] ratio in a sample of disks, and to get an overview of the strength of [CI] and warm CO emission.   We carried out a survey of the CO$\,6$--$5$ and [CI]$\,1$--$0$ and $2$--$1$ lines towards $37$ disks with APEX, and supplemented it with [CII] data from the literature. The data are interpreted using a grid of models produced with the DALI code. We also investigate how well the gas-phase carbon abundance can be determined in light of parameter uncertainties.   The CO$\,6$--$5$ line is detected in $13$ out of $33$ sources, the [CI]$\,1$--$0$ in $6$ out of $12$, and the [CI]$\,2$--$1$ in $1$ out of $33$. With deep integrations, the first unambiguous detections of [CI]~$1$--$0$ in disks are obtained, in TW~Hya and HD~100546.   Gas-phase carbon abundance reductions of a factor $5$--$10$ or more can be identified robustly based on CO and [CI] detections. The atomic carbon detection in TW~Hya confirms a factor $100$ reduction of [C]/[H]$_{\rm gas}$ in that disk, while the data are consistent with an ISM-like carbon abundance for HD~100546. In addition, BP~Tau, T~Cha, HD~139614, HD~141569, and HD~100453 are either carbon-depleted or gas-poor disks. The low [CI]~$2$--$1$ detection rates in the survey mostly reflect insufficient sensitivity to detect T~Tauri disks. The Herbig~Ae/Be disks with CO and [CII] upper limits below the models are debris disk like systems. A roughly order of magnitude increase in sensitivity compared to our survey is required to obtain useful constraints on the gas-phase [C]/[H] ratio in most of the targeted systems. 
Observations and modelling of CO and [CI] in disks. First detections of   [CI] and constraints on the carbon abundance