How can it be that the refund gets reduced by more than was added to the income? When I added $10 tips to our income then the federal refund got reduced by $13!

That can happen. I once added $6 in interest and the tax went up $12! It pushed me into the next tax bracket. I was right at the line.

plus more income can reduce any tax credits you may have.

That's the way the tax tables work. You have a tax that applies to a small range of income, sort of like an average. If you go one dollar over the highest number in the income range, your tax can jump up several dollars, as the tax is based not on the one dollar increase, but on the average of the next range of numbers. 

Thanks for the response. This is not how I understood the tax tables.

The table for 2019 shows this:

Tax rate Single filers Married filing jointly*

I would assume that a single filer would pay 10% taxes on the first 9,700 dollars and the dollar number 9,701 gets taxed with 12% and that's it. Hence If if the income would be 9,700 dollars then the single filer would need to pay 970 dollar on taxes. An income of 9,701 dollars should result in taxes of $970 plus 12 cents to a total of $970.12.

What I am missing here?

Because that is a general tax rate schedule, that is not the tax table.

See:

https://www.irs.gov/pub/irs-pdf/i1040gi.pdf  page 62 on for the tax schedule.

Yes, that is true. However, what you are using is not a tax table, it is a tax rate schedule. The tax rate schedule will give you a different tax than the tax table.

So, you can calculate your tax two different ways, and you can have two different tax amounts depending on what method you choose. The same taxable income can result in two tax amounts, both of which are accepted by the IRS.