Yes, maintain all of your records and record the details of the date of payment, name and any details you believe are relevant.

Moving expenses can be entered by going direct to the entry by using the steps here.

1. Select My Account in the upper right 
2. Select Tools 
3. Select Topic Search 
4. Type moving > Go (image attached)
5. Follow the prompts to enter your allowable moving expenses (image attached)
6. Click the blue hyper-links for more information about your specific circumstances.

Some tips about moving expense is here for your review if you choose.

Time Test:  If you are an employee, you must work full time for at least 39 weeks during the first 12 months after you arrive in the general area of your new job location (39-week test). Full-time employment depends on what is usual for your type of work in your area.  If you are self-employed, you must work full time for at least 39 weeks during the first 12 months and for a total of at least 78 weeks during the first 24 months after you arrive in the general area of your new job location (78-week test).
Distance Test :  Click the chart attached to enlarge and view.