Check your tubes as suggested above, by all means. As Bill Kapaun says above that level of leakage may or may not be reasonable depending upon the time between rides.

"Any bike shop or service station should have a tool to tighten the core."

There are also metal valve caps which have the tool built in, I get mine from my local car parts store. They will keep debris out of the valve as well as acting as as secondary seal against loose or leaky valve cores. Nice to have on a tour, I carry spares as they are easily lost.

As to your inadequate pump, check out the Topeak Road Morph G. Don't go on a tour with a known crappy pump.