CAN A PAYDAY LOAN COMPANY SUE FOR NOT PAYING YOUR WHOLE DEBT BACK

Yes, you failed to repay the loan and now the creditor will sue you for it.  Whether going to court or a settlement is better depends on what the settement is and what defenses you might have to the claim.  From your question, it doesn't appear that you have any defenses, and your loan contract probably provides that you will have to pay the lender's attorneys' fees if it sues you and wins, so a settlement is probably advisable if you can get a reasonable one.  In the absence of any defense to the claim, the only leverage you have is that if you are  unable to pay the debt you can file bankruptcy, in which case the lender will probably get less than it will if you agree to a payment schedule.  You should try to negotiate a schedule you can live with, and let the lender know that if it insists on an unworkable payment schedule, you'll have no choice but to file bankruptcy.