No. Any abilities that cost a click are actions, and actions can only be used in that players action phase, and that action must be fully resolved before another action can be used.
There are some cards where clicks are still spent during a run. The most notable card which is printed with a click costed ability is Adjusted Matrix, which modded an icebreaker into an AI with:
[Click]: Break Ice Subroutine.
This ability has been errataed to:
You may spend [Click] to break any subroutine on a piece of ice currently being encountered.
To prevent this confusion. This is the same wording as used on bioroids, such as Eli 1.0, which are the most common exception to not spending clicks outside of the action phase.