Hi,
i hope i did get your question right. What you can do is that you add an input parameter to the client actions which handles the OnClick event handler of every button you (so multiple buttons, single click handler).
The input parameter could be "ButtonPressed" with values like "ChartLeftMoreClicked" (you can also create a static entity for that).
In the client action you assign the value of ButtonPressed to a local variable within your screen.
You add an input parameter to your block, add the block to the sidebar in your screen and assign the local variable containing the button pressed value to the blocks input parameter.
You add the OnParametersChanged event handler to your block to define what happens when the input parameter changes (eg. refreshing the datasources).
You use the input parameter as a condition on various conditional statements eg "CurrentButton" = "ChartLeftMoreClicked".
Best
Stefan