Are there economic policies that help countries fight corruption? Are there particular taxes or subsidies or public goods that a country's economic authority could implement to help in the fight against corruption? While the rule of law seems to be the first order issue, and often government is where a large part of corruption exists,  it's still seems like it's  phenomenon that could be affected by economic policy.  Taxes, expenditure rules, purchasing rules, etc, are potentially useful. Maybe!
Kaushik Basu has proposed (Why, for a Class of Bribes, the Act of Giving a Bribe should  be Treated as Legal) legalizing paying bribes to separate the ties of criminality between the bribe payer and receiver. This allows the bribe payer to pay the bribe and report the bribe taker. If bribe paying is illegal then the shared criminality prevents this.
Dina Pomeranz empirically examines the widely held belief that VAT taxes discourage tax evasion (No Taxation without Information Deterrence and Self-Enforcement in the Value Added Tax) and finds that they do indeed discourage tax evasion:

Claims that the VAT facilitates tax enforcement by generating paper
trails on transactions between  firms contributed to widespread VAT
adoption worldwide, but there is  surprisingly  little  evidence. This  paper  analyzes  the  role  of  third  party  information for
VAT enforcement through two randomized experiments among over 400,000
Chilean  firms.  Announcing additional monitoring has less impact on
transactions that are subject to a paper trail, indicating the paper
trail's preventive deterrence effect. This leads to strong enforcement
spillovers up the VAT chain.  These  findings confirm that when taking
evasion into account, significant differences emerge between otherwise
equivalent forms of taxation.
Other forms of taxation that are harder to monitor and enforce provide more opportunities for corruption and so switching to a broad VAT may reduce corruption.

Another possibility is outsourcing corruption enforcement:

The International Commission against Impunity in Guatemala (Spanish:
Comisión Internacional contra la Impunidad en Guatemala, CICIG) is an
international body charged with investigating and prosecuting serious
crime in Guatemala. It was created on December 12, 2006, when the
United Nations and Guatemala, under the government of Óscar Berger,
signed a treaty-level agreement setting up CICIG as an independent
body to support the Public Prosecutor's Office (Procuraduría General
de la Nación), the National Civilian Police (Policía Nacional Civil)
and other state institutions in the investigation of a limited number
of sensitive and difficult cases. The ultimate goal is that through
CICIG's work, national judicial sector institutions will be
strengthened to continue to confront illegal groups and organized
crime in the future.

Wikipedia: International Commission Against Impunity in Guatemala (CICIG)
Civil service exams (as opposed to the spoils system which still has defenders) were, among other reasons, adopted as an anti-corruption mechanism. Perhaps these countries could also adopt such exams.

This article traces a number of cases of graft in Zambia to show the
importance of this practice within the political system. Graft is
treated as one element of a spoils system through which clientelism
operates and through which, more generally, the state is used as a
resource for private ends. Graft and patronage are shown to have
negative consequences for the state through undermining efficiency and
legitimacy and displacing policy ends. But perhaps most importantly,
it is argued that graft ultimately involves a transfer of wealth
between classes and more specifically is an important factor in the
growth of an indigenous owning class.

Political graft and the spoils system in Zambia — the state as a resource in itself
Well defined property rights provide fewer opportunities for regulatory discretion and therefore fewer opportunities for bribe takers to exert influence. Easier said that done.
Restrict financial privacy of and gift taking, side-employment, and post-employment by government employees. May be hard to enforce in low-development societies.