Yes. In California, a landlord can charge a late fee to a tenant who doesn't pay rent on time, but the landlord can do so if the lease or rental agreement contains a late fee provision. (In some communities, late fees are limited by local rent control ordinances.)
The law requires that late fees be reasonably related to the costs that your landlord faces as a result of your rent payment being late. A properly set late fee is legally valid. However, a late fee that is so high that it amounts to a penalty is not legally valid.
In your case, you have most likely violated your lease or rental agreement by not paying the full amount of rent on the first of every month. As such, the landlord can not only charge you the late fees but also serve you with a 3-day notice to pay rent or quit. The law allows your landlord to take the partial payment and still give you an eviction notice.
The information presented here is general in nature and is not intended, nor should be construed, as legal advice. This posting does not create any attorney-client relationship with the author (who is only admitted to practice law in the State of California). For specific advice about your particular situation, consult your own attorney.