In general, alimony will lower the child support you pay because it reduces your income. If you are receiving alimony and child support, the child support you receive will be lower because the alimony you also receive will increase your income which tends to lower you child support.
For example, suppose you make $80,000 per year at your job and have $20,000 per year withheld from your pay as federal income tax. Also suppose that you pay $12,000 per year in alimony.
Without the alimony payment, your gross income for child support purposes would be $60,000 per year. With the alimony payment, your gross income would be about $48,000 per year. I say “about” because, you will get a refund for the tax you had withheld on the $12,000 in alimony that you paid. Essentially, your employer will withhold taxes based on what they pay you – $80,000 per year, when in reality, you really make $68,000 (what they pay you less what you pay in alimony) – so you have too much tax withheld. The amount that you are refunded by the IRS increases your gross income.