If a person is performing tawaaf (circumabulation around the Ka'bah) or sa'i (between Safa and Marwah), and he comes across a need (e.g., he is thirsty and wants to drink something, or loses someone from his family members and stops to look for him, or becomes tired and wants to take some rest), and if the break was short, then he may continue his tawaaf from where he left off. If the prayer is called and he interrupts his tawaaf to pray, the scholars have disagreed on this issue. The most conservative opinion is that when he continues his tawaaf, he should not count the last round which he left incomplete.1
The issue of taking a rest in tawaaf and sa'i, however, is based on the condition that tawaaf and sa'i should be completed contiguously. In sa'i, continuity is not a requirement according to the best opinion.2
So, if a person is performing sa'i, and he breaks after some of the rounds, and then comes back to complete them, this would be considered permissible. However, regarding continuity of tawaaf, the scholars have two opinions:That continuity is waajib (mandatory), and that a long discontinuity without due justification nullifies the tawaaf. That continuity is a sunnah, and the tawaaf is not nullified even though the break was long.3 However, it is better to act according to the first opinion.
1 Fataawa Al-Hajj Wal 'Umrah by Ibn Baz, p. 80; Al Majmu' by Al-Nawawi, 8/49.
2 Al-Mughni, 3/414.
3 Al-Majmu' by Al-Nawawi, 8/47