When making accurate measurements of the decay of a radioactive source you must always use the corrected count rate. This is the count rate that is due to the source alone and not including any background radiation.
The idea behind this is shown in the diagram
This shows a total count rate at the start of the experiment of 120 Bq but we assume a constant background count rate of 20 Bq. This means that the count rate due to the source is 100 Bq.
Therefore if the half life of the source is 300s after this time the count rate due to the source will have fallen to 50 Bq.
This means that after the constant background count rate has been added the total activity recorded after 300 s will be 50 +20 = 70 Bq.
After 600 s the activity of the source will have fallen to 25 Bq and the total count rate recorded will now be 45 Bq and so on.