1. Others have claimed to hear a voice inside their heads that called their children’s names or had a ‘flash vision’ of what their children were doing.

有人声称能听见脑子里面有一个声音在喊她孩子的名字，或者闪现出孩子在做什么的画面。

claim payment from sb.
要求某人付款（vt. 要求、认领、索取）

Don’t claim to know what you don’t know.
不要强不知以为知。（声称、断言、主张）

No one turned up to claim on a single article.
没有人前来认领任何一件物品。(vi. 认领)

2. In any event, almost every mother seems to have it and will swear by its reliability.

无论如何，几乎每个母亲似乎都有这种感觉，并对它的存在深信不疑。

in any event ( at all events):    无论如何, 不管怎样

I will come in any event.    我无论如何要来。

swear by    对…有信心，依赖

He swears by his doctor.    他深信他的医生。