Hello, 

You could do something like this:

# get the string field in the format "mm/yyyy" and add 1 to the date
date_object = datetime.datetime.strptime("1/"+record.string_date, "%d/%m/%Y")

# add a month to the date object and minus a day to get last day of month
date_object = date_object + dateutil.relativedelta.relativedelta(months=1) + dateutil.relativedelta. relativedelta(days=-1)

id date_object < record.other_date:
    # if the date field is less than another field, do something

The idea is to get the first day of the field month, add a month to get the first day of the next month and minus a day to get the last day of the field month.

I think this would be the best way to solve this problem using the libraries supplied by Odoo using the UI. This is an example of how it would look in a server action, you just need to change the fields to the fields which are in your database table. 

I hope this helps,

Thanks,