take minimum between column value and constant global value

Use np.minimum:

In [341]:
df['MinNote'] = np.minimum(1,df['note'])
df

Out[341]:
   session      note  minValue   MinNote
0        1  0.726841  0.726841  0.726841
1        2  3.163402  3.163402  1.000000
2        3  2.844161  2.844161  1.000000
3        4       NaN       NaN       NaN

Also min doesn’t understand array-like comparisons hence your error