I read that また can mean "again", so could the phrase be written as:
(I used a casual form because I was writing for a friend)
Please let me know if this phrase would be correct! Thank you!
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'The book' refers to a book which is already known to the listener. Although Japanese does not have a direct equivalent for the word 'the', you can use the equivalent word for 'that' in this case. あの本・その本 would be 'that book'. Leaving it out would be interpreted as 'a book'.
あの本・その本を読んだ I read that book (the book which we talked about)
本を読んだ I read a book
Both を and は have been omitted from your question sentence. Without implying that this is necessarily pertinent to your case, leaving out particles before becoming proficient in their usage can cause unnecessary confusion when it comes to grammar. Even after becoming proficient, adding them back in can assist one in determining whether a sentence sounds correct.
昨日はあの本を読んだ。→ 昨日あの本読んだ。 I read that book yesterday.
昨日はあの本をまた読んだ。→ 昨日あの本また読んだ。I read that book again yesterday.
昨日はまたあの本を読んだ。→ 昨日またあの本読んだ。I read that book again yesterday.
I agree with @Kentaro Tomono about the placement of また, with the final sentence above sounding most natural.