This is a well known modeling issue and several approaches are available.
We have tried the Delta approach with good results. This is essentially a binomial glm for the presence-absence representation of the data, and conditional on the Bernoulli variable being 1, a regular continuous distribution such as Gamma glm for the positive values. Option 3) in Ben Bolker's list.
See
Aitchison, J. 1955. On the distribution of a positive random variable having a discrete
probability mass at the origin. Journal of American Statistical Association 50, 901-908
Pennington, M. 1983. Efficient estimators of abundance, for fish and plankton surveys.
Biometrics 39, 281-286
Another option is to transform your continuous response into counts. For example, if you have 0.745 kg of grass from one plot, how many stahdard 100 ml containers can you fill in a standardized manner with that? That's a count. Now if you turn your grass biomass into counts and if you are lucky (not excessive number of zeroes) then maybe a Poisson glm will be good. And the Poisson does not bug you with nuisance parameters ...
We tried several things like that, and also the Tweedie distribution (number 4) in Ben Bolker's list) in this paper:
Tascheri, R., Saavedra-Nievas, J.C., Roa-Ureta, R. 2010. Statistical models to standardize
catch rates in the multi-species trawl ?shery for Patagonian grenadier (Macruronus magellanicus)
off Southern Chile. Fisheries Research 105: 200?214
Adding a constant to the zeroes is just not right (see p. 324 of the below quoted article for an authoritative sentence on this matter):
Venables, W.N., Dichmont, C.M. 20004. GLMs, GAMs and GLMMs: an overview of theory for
applications in ?sheries research. Fisheries Research 70: 319?337.
HTH
Rub?n