1. What does it mean?
생카 shortens 생일 카페, birthday cafe. Fans arrange a themed cafe event to celebrate a favorite person's birthday, often with displays or special items. The person being celebrated does not have to be present.
Also written or said as 생일카페, saengka
2. The nuance
It is not simply a cafe serving birthday cake. Many events are fan-organized; attendance by the artist or official endorsement should never be assumed.
3. Examples
“최애 생카에 가서 장식 사진을 찍었어.”
I went to my favorite idol's birthday cafe event and photographed the decorations.
“생카 운영 시간이 평소 카페 시간과 다르대.”
Apparently the birthday event hours differ from the cafe's normal hours.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.