If I missed a night of sefirah, and counted the next day exactly 10 minutes after shkiah, can I continue to count with a bracha the next night?
You can continue to count with a berachah.
This is a difficult question, and there are those who write that one cannot continue counting with a berachah. The reason for this is if one counts in the day one can continue counting on account of a safek sefeika – it is possible that one can count in the day, and even according to the opinion that one may not count in the day, it is possible that one can count even after missing a day.
The Rashba writes that a safek sefeika must involve two full sefeikos, and in the case of counting after shekiya the second safek is not a full safek, because of the additional consideration that perhaps it is already night. This rationale is stated by the Beis David (268).
However, it is possible that the reason why one can continue to count is not because of a safek sefeika, but rather because of a mima nafshach: According to the opinion that counting by day isn’t good, one can continue even if one misses a day, and according to the opinion that missing a day is a problem, counting by day is fine. According to all opinions it therefore emerges that if one counts in the day one can continue counting (Maamar Mordechai 489).
Based on this, it follows that if one counts a few minutes after shekiya (in bein ha-shmashos) there will be a safek sefeika: If the time is day, everything is fine, and even if the time is night there is an opinion that missing a day doesn’t stop the count.
In addition to this, the Sha’ul U-Meishiv writes that even based on the safek sefeika rationale, counting in bein ha-shmashos is sufficient, because counting is specifically “definite” and the time for counting only begins at nightfall, and Shut Beis Shlomo (102) likewise writes that one can continue to count on following days.
Therefore, the general consensus of poskim is that one can continue to count with a berachah, and this can be relied on in practice (see also Mitzvas Sefiras Ha-Omer 35:5).