Moving to Apartments

eliminating "to" better describes that it is someone else that is moving, and they are moving from one apartment to another apartment. Alternatively, "Moving to a New Apartment" could still keep the word "to" but fixes the awkwardness of a missing indefinite article (a/an).

Yesterday, my friend called meto asking me to help a lady who is moving home, whouse because she has just divorced her husband.

"asking me" is correct in speech, but "to ask me" feels more correct in writing.
"moving home" usually means moving back home after living elsewhere for awhile. "Moving house" means moving from any house to another house.

She wais planning to move from a free, government-funded apartment to a new apartment where she will needed to pay $800 a month, andfter the government paid anotherys $600.

"Was planning" would indicate that the move did not happen, but "government paid" indicates that it already did happen. Making the tenses consistently in the present continuous tense helps explain that this is a hypothetical plan that is happening in the future.

Moving to Apartment

Yesterday, my friend called me asking me to help a lady moving home, who has just divorced her husband to move home.

She was planning to move from a free government-funded apartment to a new apartment where she needed to pay $800 a month, and the government paid another $600.

Moving to Aan appartment

Yesterday, my friend called me to asking me to help a lady moving home, who has justwho has recently divorced her husband., move houses.

She was planning to move from a free government-funded apartment to a new appartment where she needed to pay $800 a month, and the government paid ano further $600 towards the rent.