McLeish (1974) (see http://projecteuclid.org/download/pdf_1/euclid.aop/1176996608) proofs in his paper a CLT for martingale difference arrays, but I am a little bit confused by the assumptions. At page 1 he writes "...finite second moments are generally not assumed ...". But one of the assumption is that $\max_{i\leq k_n}|X_{ni}|$ is uniformly bounded in $L^2$ (i.e. $\sup_{n\geq 1}\mathbb{E}[|\max_{i\leq k_n}|X_{ni}||^2]<K<\infty$).
Also, in this book Theorem 5.2.3 (see https://books.google.at/books?id=Sqg-YPcpzLYC&pg=PA36&lpg=PA36&dq=flemming+counting+uniform+integrability&source=bl&ots=T1mhlvKVzu&sig=Su-fuZeO0ZcbLuqRXluvH_4B-CY&hl=de&sa=X&ved=0ahUKEwj3qsjrwZXVAhXLiRoKHXsWB2gQ6AEIMTAB#v=onepage&q=flemming%20counting%20uniform%20integrability&f=false) the author writes (one paragraph over the Theorem, which cannot be seen in the preview): "...we want a result without finite variances ... a result due to McLeish provides the basis for the necessary generalization"
Doesn't follow from the mentioned uniform boundedness that the variances are finite??
I think the condition that bothers you is in Theorem 2.3, and is remedied by Corollary 2.7. I think (and I don't want to read this dense paper closely, so I'm not 100% sure) that in situations like Cor. 2.7 you can apply 2.3 to the truncated summands. In other words, 2.3 is a tool, and not the main result. The author could have been a little more helpful in providing a roadmap, or in phrasing his results better.