Sorry, but this is a limitation of HPE-Synchronizer. It does not work with that field at all.
I suppose you could use QC workflow to copy the current value for Target cycle to a STRING custom field, then map that field to JIRA, but it would not be able to be bi-directional, or at least would be a ''suggested'' value for someone to set interactively in QC.
+++++++++++++++++FROM the README:
The following ALM fields are not available for mapping and are not displayed in the list of fields available for mapping between the endpoints:
Defect entity fields: Subject.
Requirement entity fields: Target Cycle.
This limitation applies when a field is mapped from ALM to an endpoint that does not have multibyte character support. If the field's value for a record in ALM is longer than the maximum length of the field in the other endpoint and contains non-English characters, the Synchronizer is unable to synchronize the record and writes an error message to the log file.