Even though I have very very little knowledge of the mechanics of an automobile I think I have a gut feeling the hesitation has to due with an electrical problem I have had since I bought the ute. After about a week of driving or so, my battery goes flat and I have to recharge it for the next week. If I leave it in the garage without driving it the battery goes flat even sooner, about two days. For example, I haven't driven it the past 3 days and the battery went from normal (approx. 12v) down to (5v) after measuring it with my multimeter.

That being the case..totally speculating here but..The hesitation could be due to low voltage in the battery? Then after 10 minutes or so of driving the alternator brings the battery back up to its normal range and that's why the problem goes away? Could this be the root of the problem?