Your AccuCheck knows what your blood sugar levels are for just those five times you did a finger prick and has no idea of how your blood sugars are doing for the rest of the day. It is effectively guessing without 90-99% of the the data it needs. If your blood sugars levels are relatively steady then the estimate may be accurate, but if they are not, then the estimate could be totally incorrect.
It is unlikely the hospital test is a snapshot,(unless they just used a finger prick test) and if the hospital drew blood using a needle, then they would be doing a proper HbA1c test which is not a snapshot, but shows an accurate result for the last 90 days.
To give you an example, when I was doing finger prick readings, I was in normally in range before I went to bed and when I got up in the morning. When I got my freestyle libre, I could see that there was one particular meal that was causing my blood sugar levels to starting rising four hours after eating. This meal was something I tended to eat when I was having a late dinner and hence the four hour rise started after I had gone to bed and hence after the last finger prick. The consequence was that my sugar levels where high for most of the night without me realising that. If that had of been a regular meal it would have meant that my HbA1c was much higher than the finger pricks indicated.