Someone who behaves oddly, as a sick puppy might; for example, a lovesick person who pines after their beloved.
This phrase originated in America in the early 20th century. There are many examples in print of people being described as 'like a sick puppy' - these usually relate to men mooning around after women they are attracted to; for example, this piece from The Indianapolis Star, 7th May 1911:
"When a noted actress is in town," said one detective yesterday, "lots of times some poor fool, wearing a carnation in his coat lapel, will whine around after her like a sick puppy."
It took some time for the metaphor to develop. The first use I can find of something that is described as being a 'sick puppy', as opposed to 'like a sick puppy' is in a report in the Oklahoma newspaper The Lawton Constitution And Morning Press, 29th December 1957. In this piece the wealthy industrialist, and entrepreneur Powel Crosley Jr. talks about his purchase of the baseball team The Cincinnati Redlegs and he uses 'sick puppy' to describe the club rather than an individual:
"I took the club years ago when it was a sick puppy. It was about $200,000 in debt.