When one goes into the nursing home their income goes with them to help pay; Medicaid pays the balance of the bill. The $35 is a personal allowance residents are allowed for things like beauty salon bills, etc
not a rule that they have to do it this way but a very common practice. you will not easily find a nursing home that won't have someone sign over their checks.
Medicaid is the payor of last resort. This means that Medicaid only pays the difference between a person's income and the cost of nursing home care. The person receiving Medicaid is required by law to make a co-payment equal to their income, minus a small personal needs allowance (the $35). In some states the person also has to pay their Medicare supplement insurance premium aout of their co-payment portion.
In Florida it is called patient responsibility. The state through the institutional care program pays the balance, after the resident pays their income minus $35.00 to the facility. It can be calculated differently if there is a community spouse (spouse living outside the facility). You may want to meet with an elder law attorney in your area to explain the Medicaid assistance program your mother is using.