Start with the burner cap. Let everything cool first, then lift the cap and set it back so it locates in the base. The CornuFe manual says heads must seat correctly, with two pips fitting two notches, and that parts must be completely dry before they go back. Damp parts can leave a burner clicking without lighting after a cleaning.
Then look for blockage. The cleaning tips say to wash gas burner caps in warm soapy water and make sure every hole around the cap is free of grease and debris. The manual says blocked ports can be cleared with fuse wire, but use a light touch and never work around a burner while another burner or the oven is on, because the ignition system can shock you.
Last, confirm the power is on. The cooktop needs electricity to spark, so a tripped breaker or a power cut will leave you with no click. During a power failure, the CornuFe manual describes lighting a cooktop burner with a match, while the Chateau gas oven manual prohibits lighting the oven with a match. Read the exact steps in your own manual before trying either.