If the sentence had been
I saw a love (of mine) in the street
That would have been fine. It would mean I saw an ex-lover walking down the street.
The expression a love refers to a lover
Oxford Dictionaries defines the noun love as
3 [COUNT NOUN] A person or thing that one loves.
- ‘she was the love of his life’
- ‘their two great loves are tobacco and whisky’
Thomas Hardy's The Sacrilege contains the following lines
"I have a Love I love too well
Where Dunkery frowns on Exon Moor;
I have a Love I love too well,
To whom, ere she was mine,
'Such is my love for you,' I said,
'That you shall have to hood your head
A silken kerchief crimson-red,
Wove finest of the fine.'
Note that Love is spelled with a capital letter. It is a term of endearment, a name to call your spouse, or sweetheart.
Unfortunately, I saw a love on the desk makes little sense to me, it sounds as if a person saw their lover standing on top of a desk; it's not physically impossible, but it is highly unlikely.
A more plausible sentence would be:
I saw a silver framed photo of a loved one on a desk.