Can I use my temporary US Passport to leave/re-enter the US?

No

Your temporary passport can only be used to enter the US one time.

Before you depart the US again you must get a regular passport

Return to US on the short-term passport. Then apply for a full passport before attempting to travel anywhere else.

IF you must travel in short order, e.g. business travel for your employer, then follow directions to apply to use expedited service at the nearest Passport Agency. There is at least one in each state. Make an appointment by phone, appear in person with all your documentation, application & fees. Your new passport will be issued no later than the following day (if you take a late afternoon appointment). If you do not qualify to use the Passport Agency, expedited service takes about 2 weeks, 3 at most during the heavy travel season (like right now). Having used both, expedited service took 10 calendar (6 biz) days, & Passport Agency took just under 3 hours.

You can get a brand spanking new US passport within about 180 minutes from the next passport agency.

What are you talking about?

you have to return to ther uS first ..then renew your passport before the next travel

This is a valid concern that you should take very seriously. Because of visa time limits, many countries require your passport to be valid for 3-6 months beyond your departure date (regardless of when you would actually return).

In a nutshell, this means if you travel to those countries and your passport expires in a month, it might not be valid and you may not be able to board a flight.

You should explore options to get your permanent passport from Europe while you have time if your passport will expire in less than 6 months when you board your Caribbean flight.