But 'the year' means a particular/specified year or the one which is. In that full version, each occurrence of the word year. The year in review could be used as the name of this year.
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'a year' can be any year without any specification. Unless you're talking about chinese (or persian) new year, the name of the celebration isn't new year but new year's eve, and it happens. The comparison with the second year and the last year supports plural rather than singular.
I'd say 'across the year' hints strongly at sporadic events etc, whereas 'throughout the year' speaks more of consistent results.
