Isaiah 51:22 (King James Bible) thus saith thy Lord the LORD and thy God that pleadeth the cause of his people behold …
The printer’s manuscript shows an initial copy error that seems to be based on the preceding use of thy in “thy Lord” and “thy God”. The manuscript correction of the thy to his is in heavier ink flow, which suggests that the change was not immediate (but it could have been made later while proofing 𝓟 against 𝓞). The original manuscript is not extant here, but it probably read his. The correction in 𝓟 makes the Book of Mormon text agree with the King James text of Isaiah, yet there is no clear evidence that a King James Bible was used by the scribes to proof either of the two Book of Mormon manuscripts.
Errors in selecting the correct possessive pronoun are quite common in the Book of Mormon manuscripts, as can be seen with the examples listed under 2 Nephi 3:12, 14. In a nearby example here in chapter 8, Oliver Cowdery initially copied mine as thy in 𝓟 but then immediately corrected his error (see 2 Nephi 8:16). For that example, the original manuscript is extant.
Summary: Maintain in 2 Nephi 8:22 the corrected text in 𝓟 (“the cause of his people”); Oliver Cowdery initially wrote “the cause of thy people”, an error apparently deriving from the occurrence of “thy Lord” and “thy God” earlier in the sentence.