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.)

<item>
    <name>Mouse Warping</name>
    <vkopenurldef>
      <name>KeyCode::VK_OPEN_URL_SHELL_mousetowindowcenter</name>
      <url type="shell">
        <![CDATA[
          /Applications/Karabiner.app/Contents/Library/utilities/bin/warp-mouse-cursor-position front_window middle 0 center 0
        ]]>
      </url>
    </vkopenurldef>
    <item>
        <identifier>private.f14_mousetowindowcenter</identifier>
        <name>F14 to move the mouse to the center of the current window</name>
        <autogen>
            --KeyToKey--
            KeyCode::F14,
            KeyCode::VK_OPEN_URL_SHELL_mousetowindowcenter
        </autogen>
    </item>
</item>

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.