FLK is a politically incorrect and slightly humorous reference to a 'Funny Looking Kid'. Despite the obvious negative connotations, the label FLK is a useful, if informal, diagnostic indicator, and it is sometimes used by doctors and therapists to refer to their clients. While there are any number of very specific diagnostic indicators that can alert you to possible health issues, often the first indication that something is wrong is the impression that 'this kid looks odd'.
While you can never assume that someone who looks odd has any sort of problem, nor that someone who looks normal is normal, an infant or child who 'looks funny' is worth a second look. Slight cranial or facial deformities, unusual carriage and posture, and abnormal social behavior all may be the result of deeper structural, functional, or cognitive issues. These factors are not always blatantly obvious, but we are very attuned to how humans naturally act and look, and we may sense abnormalities that are not apparent to initial formal screenings.
Obviously, a parent should never hear a doctor or therapist refer to a child as an FLK, and many professionals will avoid this initialism like the plague. However, the information conveyed, that something seems wrong but you're not sure what, is valid and often important. If nothing else, it is important for any professional to recognize that 'feeling that something is odd' is a common and professionally relevant experience.
The facetiousness of the label FLK should be taken as an indication that the professional is acknowledging that their feelings are far from objective, and not as mocking the child in question. However, FLK is certainly common knowledge in the medical and allied health fields, and is not likely to disappear anytime soon.