By default Tooltip is shown with top-left at cursor.
This way it obscures underlying Control and mouse clicks are not delivered without further explicit mouse movement.
I strongly suggest to displace Tooltip a few pixels to avoid problem.
This was already discussed in {html}<A href="https://javafx-jira.kenai.com/browse/RT-28269">RT-28269</a>{html} and then deferred to "more general" {html}<A href="https://javafx-jira.kenai.com/browse/RT-33093">RT-33093</a>{html}. Unfortunately the latter was closed "Won't fix".
Proposed workaround should be quite painless and resolve the basic misbehavior without waiting for heavier (and sometimes unwanted!) {html}<A href="https://javafx-jira.kenai.com/browse/RT-33094">RT-33094</a>{html}.
Please consider implementing it as default behavior.