-Believe it an engine problem, not a pump problem.

-Sounds like a fuel problem to me, either too much or too little.

-Agree; if this is new, take it back/discuss with vendor.

-If no warranty, used, or just too big a hassle, then is the dying accompanied by black smoke from exhaust?

if yes, then suspect too little air and agree with others choke or air cleaner.

if no smoke then suspect too little fuel; check fuel filter isn't clogged, fuel lines sucking air thru small leak, junk in carb float bowl, etc. Also possibly vapor lock; check make sure fuel lines not routed too close to exhaust pipng, exhaust discharge, or heads, check cooling air ducting intact, etc.