I know it's obvious but before you can make any fix, you need to find the cause of the movement and prevent it from continuing. If you don't solve this first, no matter what you do, the CONTINUED movement will have you making a 'fix' every year for as long as you fix the house.
Here's a Post and Follow-up where a Homeowner is experiencing a similiar problem. Again, the cause can be a number of things so read my Follow-up at least TWICE so you understand the importance of the Gutter and Downspout 'issue'. This is the MOST common cause (and related contributors) to water in the basement and foundation problems. Sure, your cause may be from, say ..., tree uprooting, but the point is the same - Fix the cause BEFORE you fix the problem.
Hole In Foundation In your case, an expansion / hydraulic cement should be OK. I really can't see the problem from here so at least consider it. (Perhaps a new walkway is the answer???)