▲你的判断是有道理的，通常的用法是 have a bearing on（与……有关，对……有影响）。如：
What he said had not much bearing on the problem. 他说的话跟这个问题没有多大关系。
The rise in interest rates had a direct bearing on the company’s profits. 利率的提高对公司的利润有直接的影响。
The case has no direct bearing on the issues being considered. 这个案例与正在考虑的问题没有直接联系。
This material is of such importance that it has a powerful bearing on the case. 这个材料太重要了，与这个案子密切相关。
Experts generally agree that diet has an important bearing on your general health. 专家们普遍认为饮食和健康总体状况密切相关。
My father’s achievements really don’t have any bearing on what I do. 我父亲的成就与我做的事情真的毫无关联。
▲很少见到表示以上用法时用 have a bearing with 的。我查了手头大大小小十多本词典也没有查到，在BNC（英国国家语料库）也没有查到。所以估计你的第一句中用介词 with 是个失误。