Can I add multiple sources of passive foreign taxes from one country in one entry on form 1116 or should I enter them one at a time?

In order to claim a foreign tax credit, in TurboTax, and via IRS Form 1116, you do need to specify some country (or countries) as the source of your foreign taxes paid / accrued.

However, the country selection "Various" is available in the drop-down country menu, as is the three-letter acronym "RIC" which stands for "Regulated Investment Company" (which is basically another term, or pseudonym, for a mutual fund or funds).  Thus, you could use either "Various" or "RIC" as your choice to resolve the issue of multiple sources of passive foreign income.

It would also be acceptable to combined the passive income from several different sources (brokerage accounts, for example) into a single entry from a specific country.  For example, you could add together brokerage accounts A, B, and C, all reporting passive income from the United Kingdom, into a single entry.

The only thing here to be especially mindful of is mixing different kinds of foreign income taxes paid.  For example, and sticking with the United Kingdom simply for sake of example, it would never be proper to mix foreign taxes paid from passive sources and active (e.g. wage) sources, even if they are from the same "foreign" country (U.K.).