10 minutes back when I checked the score, X team was losing.
Is this expression correct? Use of the word 'when' sounds odd to me. What would the right construct be?
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Yes, when is perfectly fine in that sentence.
You are saying that, in the moment you checked the score, it was unfavorable for that team. Five minutes earlier, the score could have been different, but you don't know it; the score could have changed five minutes after you checked, but you don't know it.
I would rather rephrase the sentence as:
10 minutes ago, when I checked the score, X team was losing.
When I checked the score, 10 minutes ago, X team was losing.
This is correct. The phrase with when is a distinct modifier and can be moved or omitted; and when you write it, it might be set off with a pair of commas :
10 minutes back, X was losing.
When I checked the score, X was losing.
10 minutes back, when I checked the score, X was losing.
As tchrist tells you, ago may be substituted for back; but this is true only if your 'reference time', the time to which you are relating the event, is the present moment when you are speaking. If you are relating the event to other events in the past you cannot use ago but must use before that or back:
Right now X is winning; but 10 minutes ago when I checked the score, X was losing.
At half time X was winning; but 10 minutes back, when I checked the score, X was [or, had been] losing.
Note that this use is different from back when, an intensive variant of when which locates the event you are speaking of in an explicitly remote past:
Back when I was a boy, Alabama schools were still segregated.
All of these uses of back are colloquial rather than formal.
OP's usage is "correct", in that I doubt anyone can cite a "grammatical rule" forbidding it, but...
...I'd advise against it, purely on the ground virtually no-one else uses the form.
As StoneyB points out, "ago" is only valid when the "reference time" is the current time of speaking, whereas "back" can in principle also be used of a reference time in the past...
This afternoon he scored a brilliant goal! He'd been on the substitute bench only 5 minutes back.
But I don't like that one little bit. I'd recommend using ago whenever the referrence time is the present, and earlier, before, prior to that, or previously, for example, if the timeframe is entirely in the past.