The race for Rashtrapati Bhavan seems to be in the final leg with a relay race of sorts on between Congress and its allies. And now even the Opposition is being wooed to join in.
Please explain the highlighted ones.
Relay races have four legs, each one run by a different runner. The elections are now entering their final phase [the final leg].
I am not sure quite what the author means by "a relay race of sorts between Congress and its allies" - I think he simply means they are in competition with each other.