In the printer’s manuscript, Oliver Cowdery initially wrote “and began to put heavy burdens upon them”; then he crossed out the them and supralinearly inserted their and wrote backs inline at the beginning of the next line. Thus the correction was clearly immediate; undoubtedly the original manuscript read their backs. Oliver probably wrote them because he had just written them in the preceding predicate (“and exercise authority over them”); there are also two other occurrences of them in this long sentence (“they would smite them on their cheeks … and drive them as they would a dumb ass”).
Summary: Maintain the immediately corrected reading in 𝓟: “and began to put heavy burdens upon their backs”.