"next week": these are fine, ne[ks] week/day, ne[kst] week/day
"left side": these are fine, le[f] side, le[ft] side.
Here's the reason speakers don't always pronounce [t]. The letters <xt>, for example, in the word next are pronounced [kst], that's three sounds. Three sounds take more time and energy to pronounce than two sounds, so speakers drop the last sound, [t].
"one more": these are fine, o[m]e more, o[n]e more
The sound change above has to do with anticipation. Both [n] and [m] share the same manner of articulation; they are nasals. The speaker in anticipation of the <m> in more pronounces <n> as [m], giving o[m]e more.
Hope that helps.