The XXL Survey L. Active galactic nucleus contamination in galaxy clusters: Detection and cosmological impact

X-ray observations of galaxy clusters are impacted by the presence of active galactic nuclei (AGNs) in a manner that is challenging to quantify, leading to biases in the detection and measurement of cluster properties for both astrophysics and cosmological applications. We detect and characterise clusters contaminated by central AGNs within the XXL survey footprint and provide a systematic assessment of the cosmological impact of such systems in X-ray cluster samples. We introduce a new automated class for AGN-contaminated (AC) clusters in the XXL source detection pipeline. The majority of these systems are otherwise missed by current X-ray cluster-detection methods. The AC selection is also effective in distinguishing AGN and cool-core presence using supplementary optical and infrared information. We present 33 AC objects, including 25 clusters in the redshift range, $0.14 \leq z \leq 1.03$, and eight other sources with significantly peaked central profiles. Six of these are new confirmed clusters. We computed the missed fraction of the XXL survey, which is defined as the fraction of genuine clusters that are undetected due to their centrally peaked X-ray profiles. We report seven undetected AC clusters above $z > 0.6$, in the range where X-ray cluster detection efficiency drops significantly. The missed fraction is estimated to be at the level of $5\%$ for the 50 square-degree XXL area. The impact on cosmological estimates from missed clusters is negligible for XXL, but it produces a tension of $\sim 3\sigma$ with the fiducial cosmology when considering larger survey areas. Looking towards surveys such as eROSITA and \textit{Athena}, larger areas and increased sensitivity will significantly enhance cluster detection, and therefore robust methods for characterising AGN contamination will be crucial for precise cluster cosmology, particularly in the redshift $z > 1$ regime.