The only non-primate animal that we know of to have sex purely for enjoyment are some of the dolphin species.
Wood pigeons, like most birds, annually rear more young than the environment can support. This is a specific insurance against times of famine and disease and importantly , provides a surplus of birds for the next breeding season.
To ensure the fittest birds breed on to perpetuate the species, the birds rebond their pairing many times a day by going through the display-feed-mate process. If one of the pair is not in good condition and lacks in this rebonding the other one will soon loose interest and become interested in another prospective mate.
Most breeding birds are only receptive for a short time after the egg is released from the ovary. The other times that mating takes place are for pair-bonding purposes and often no sperm is transferred if there is a fertilised egg coming down the ovary - lack of hormones that activate certain muscles in the hen's cloaca make transfer impossible.
Saying that, a few bird species have special adaptations where the hen bird, especially in those species that use a lek for display, are able to store viable sperm and fertilise the whole clutch of eggs from one mating.