You can set the mini menu to appear when hovering a mouse pointer over an AddToAny button.

var a2a_config = a2a_config || {};
a2a_config.onclick = false;

In Drupal, add the following JavaScript code to your "Additional JavaScript" box in Configuration > Web services > AddToAny > Additional Options.

In WordPress, add the following JavaScript code to your "Additional JavaScript" box in Settings > AddToAny.

a2a_config.onclick = false;

Additionally, you can set a hover delay.

Instead of display on hover, you can set the mini menu to display on click.