Jump the pointer long distances with keyboard?
Karabiner
Karabiner comes with everything you need—a way to define a shortcut, and a way to move the mouse. The warp-mouse-cursor-position utility lives in /Applications/Karabiner.app/Contents/Library/utilities/bin/.
It has a default setting to warp the mouse to the center of the current window by tapping fn or holding ' or /, but for a customized version, add this to your ~/Library/Application Support/Karabiner/private.xml: (After you’re done, go into Karabiner’s preferences and check the box inside Mouse Warping.)
&lt;item&gt;
    &lt;name&gt;Mouse Warping&lt;/name&gt;
    &lt;vkopenurldef&gt;
      &lt;name&gt;KeyCode::VK_OPEN_URL_SHELL_mousetowindowcenter&lt;/name&gt;
      &lt;url type=&quot;shell&quot;&gt;
        &lt;![CDATA[
          /Applications/Karabiner.app/Contents/Library/utilities/bin/warp-mouse-cursor-position front_window middle 0 center 0
        ]]&gt;
      &lt;/url&gt;
    &lt;/vkopenurldef&gt;
    &lt;item&gt;
        &lt;identifier&gt;private.f14_mousetowindowcenter&lt;/identifier&gt;
        &lt;name&gt;F14 to move the mouse to the center of the current window&lt;/name&gt;
        &lt;autogen&gt;
            --KeyToKey--
            KeyCode::F14,
            KeyCode::VK_OPEN_URL_SHELL_mousetowindowcenter
        &lt;/autogen&gt;
    &lt;/item&gt;
&lt;/item&gt;

With Karabiner's utility, you can use the screen option like this:
warp-mouse-cursor-position screen NUM VERTICAL X_OFFSET HORIZONTAL Y_OFFSET

VERTICAL can be top, middle, or bottom, and HORIZONTAL can be left, center, or right.
X_OFFSET and Y_OFFSET can be numerical values, like +10 or -10. Just be careful to put the arguments in the right order—they seem oddly switched.
(For the front window, just use front_window instead of screen.)
cliclick
Probably the most popular way, but in order to use it for moving to the middle of a window, you’d need an AppleScript to get the bounds of the current window, then calculate the center, etc.