Context. The XMM-Newton survey of the Small Magellanic Cloud (SMC) revealed 3053 X-ray sources with the majority expected to be active galactic nuclei (AGN) behind the SMC. However, the high stellar density in this field often does not allow assigning unique optical counterparts and hinders source classification. On the other hand, the association of X-ray point sources with radio emission can be used to select background AGN with high confidence, and to constrain other object classes like pulsar wind nebula. Aims. To classify X-ray and radio sources, we use clear correlations of X-ray sources found in the XMM-Newton survey with radio-continuum sources detected with ATCA and MOST. Methods. Deep radio-continuum images were searched for correlations with X-ray sources of the XMM-Newton SMC-survey point-source catalogue as well as galaxy clusters seen with extended X-ray emission. Results. Eighty eight discrete radio sources were found in common with the X-ray point-source catalogue in addition to six correlations with extended X-ray sources. One source is identified as a Galactic star and eight as galaxies. Eight radio sources likely originate in AGN that are associated with clusters of galaxies seen in X-rays. One source is a pulsar wind nebula candidate. We obtain 43 new candidates for background sources located behind the SMC. A total of 24 X-ray sources show jet-like radio structures.