Can I evict with partial rent payment?

[Brian Zaug]

So this duplex I have is rented with section 8 tenants.   I  received a call the other day, as well as a letter,  from the Sect. 8 dept. saying that my tenants voucher has been reduced from $1100 (full rent) to $800 due to the tenant getting a job.  So now the Tenant is responsible for 300 a month.  Her and I both received this letter and call.  A day later, my tenant called me saying that the job was temporary and that she will be receiving the full rent reimbursement again.  I called the sect. 8 office to confirm this and they said that she never handed in the paperwork necessary for this to readjust back to the full rent reimbursement.  This is typical of this tenant. 

 I called her 2 days ago to give her the heads up because rent will be due soon and she already says she doesn't have the money.  I do have a grace period in the lease of 4 days so if she doesn't pay her portion ($300) of the rent by then, I'll add the late fee but could I also hit her with the Pay or Quit notice?   The $800 from sect. 8 gets direct deposited on the 1st. 

 Could I accept the $800 and still file for eviction due to her not paying her part ($300)?  

 Is there a better route I should take? 

She's been an ok tenant.  Would take a good amount of work/money to clean that apartment up so I'd prefer not to kick her out...

Thanks in advance...