If the Broken Time-Space stadium is in play, can I use Rare Candy on a Basic Pokemon that just came into play this turn?
No, the wording of Rare Candy specifically says "you can't use this card during your first turn or on a Basic Pokémon that was put into play this turn". Broken Time-Space does not allow you to override that.
Source: PUI Rules Team (2011-06-09)

Relevant cards

Note: All applicable cards may not be listed below.
BW10_105
BW10_85
BW5_100
DP4_102
EX13_90
EX2_88
EX9_83
HGSS2_82
POP5_7
POP8_10
SM1_129
SM2_165
SM7_142
SWSH1_180
SWSH1_180
XY5_135