Let $X$ be a projective 3-fold in characteristic $p>0$. Let $(X, D)$ be a klt pair, and $D'$ a $\mathbb{R}$-Cartier divisor such that $D'=A'+B'$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor and $B\geq 0$ is a $\mathbb{R}$-divisor. Further assume that $(X, D+B')$ is klt and $(X, D+D')$ is log canonical but not klt. Let $Z\subset X$ be a minimal log canonical center of $(X, D+D')$.

Then show that there exists an effective $\mathbb{R}$-divisor $D''\sim_{\mathbb{R}}D'$ such that $(X, D+D'')$ is log canonical and $Z$ is the only log canonical center, and there is only one exceptional divisor $E$ over $X$ such that the discrepancy $a(E; X, D+D'')=-1$.

Observer that most of what I asked in the statement above is a part of the standard tie-breaking trick, except the part where I want the new **$D''$ to be ($\mathbb{R}$-) linearly equivalent to $D'$**. This can be arranged easily in characteristic 0 using Bertini's theorem for base-point free linear system, which fails in positive characteristic. Let me explain below how does the proof of the above statement work in char 0 so that I can pinpoint the difficulty in char p>0.

By the standard Tie-breaking trick, for example, see Proposition 8.7.1, page 152 of the book "Flips for 3-folds and 4-folds" by Allesio Corti, there exist an ample $\mathbb{Q}$-divisor $A''\sim_{\mathbb{Q}} A'$, and $0<\varepsilon, \eta\ll 1$ such that $(X, D+B'+(1-\varepsilon)A'+\eta A'')$ is log canonical with $Z\subset X$ as the only log canonical center and also has a unique log canonical place over $Z$. This much is true over all characterisitc, including char p>0, as long as the existence of resolution of singularities is known.

Now let $f:Y\to X$ be a log resolution of $(X, D+B'+(1-\varepsilon)A'+\eta A'')$. Then there is only one exceptional divisor, say $E_0$ with discrepancy $-1$. Since $\varepsilon>0$ and $\eta>0$ are sufficiently small we may assume that $0<1-\varepsilon+\eta<1$. Choose $0<\delta<1$ such that $1-\varepsilon+\eta+\delta=1$.

Now in char 0 by Bertini's theorem for the base-point free linear system $|f^*A'|_{\mathbb{Q}}$, there exists an ample $\mathbb{Q}$-divisor $H'\sim_{\mathbb{R}} A'$ such that $f^*H'$ has smooth support, $f^*H'=f^{-1}_*H'$, and that $f^*H'$ intersect the exceptional divisors of $f$ and the strict transfrom of other divisors tranversally. Then $(X, D+D'')$, where $D''=B+(1-\varepsilon)A'+\eta A''+\delta H'\sim_{\mathbb{R}} B+A'=D'$, is log canonical with $Z$ as the only log canonical center and has a unique log canonical place over $Z$.

So in char p>0, without the Bertini's theorem for base-point free linear system I don't know how to complete the argument. I am wondering if there is any **$F$-singularity version of Tie-breaking which could help here**. More specifically, I am wondering if the following statement is true:

Let $(X, D)$ be a simple normal crossing pair, i.e., $X$ is smooth and $D$ has simple normal crossing support. Let $B\geq 0$ be an effective Cartier divisor such that the linear system $|B|$ is base-point free. Then there exists a $0\leq B'\sim_{\mathbb{Q}} B$ such that a closed subset $W\subset X$ is a **sharply $F$-pure center of $(X, D+B')$ if and only if it is a sharply $F$-pure center of $(X, D)$**.