the short answer
Lengthened partials are partial reps performed on purpose in the stretched half of a lift, the deep bottom portion where the working muscle is at its longest, rather than the lockout or squeeze at the top. Picture the bottom third of a dumbbell curl with the arm hanging long, the deep stretch of a cable fly, or the low position of an overhead triceps extension: instead of completing the whole range, the client works short, controlled reps in that stretched zone. They are also called long-length partials or stretch-position reps, and they are usually added on top of normal full-range work, not used in place of it. That makes them the opposite of the old-school lockout partials, which trained only the top, shortened end of a movement.
This article is general training education, not medical advice. The technique is low risk on the right exercises but a poor idea on the wrong ones. If a client is managing a joint or tendon problem, injury or health condition, route them to a physiotherapist or doctor rather than experimenting with load in a stretched position.